Maths4U Original · Short practice

Let \(\displaystyle a=2\). Give exact values.

Evaluate \(a^{2}a^{1}\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle a=3\). Give exact values.

Evaluate \(a^{3}a^{2}\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle a=3\). Give exact values.

Evaluate \(\displaystyle \frac{a^{3}}{a^{5}}\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle a=4\). Give exact values.

Evaluate \(\displaystyle \frac{a^{4}}{a^{7}}\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle a=4\). Give exact values.

Evaluate \((a^{4})^{3}\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle a=5\). Give exact values.

Evaluate \((a^{5})^{1}\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle a=5\). Give exact values.

Evaluate \(a^0a^{1}\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle a=2\). Give exact values.

Evaluate \(a^0a^{2}\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle N=729\).

Evaluate \(\displaystyle N^{\frac{1}{2}}\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle N=4096\).

Evaluate \(\displaystyle N^{\frac{1}{2}}\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle N=4096\).

Evaluate \(\displaystyle N^{\frac{1}{3}}\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle N=15625\).

Evaluate \(\displaystyle N^{\frac{1}{3}}\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle N=15625\).

Evaluate \(\displaystyle N^{\frac{2}{3}}\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle N=46656\).

Evaluate \(\displaystyle N^{\frac{2}{3}}\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle N=46656\).

Evaluate \(\displaystyle N^{- \frac{1}{2}}\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle N=117649\).

Evaluate \(\displaystyle N^{- \frac{1}{2}}\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle N=\)\(6400000\). Here a and n denote the standard-form coefficient and exponent, not given constants.

Write \(\displaystyle N=a\times 10^{n}\) in standard form. Find a.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle N=\)\(65000000\). Here a and n denote the standard-form coefficient and exponent, not given constants.

Write \(\displaystyle N=a\times 10^{n}\) in standard form. Find a.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle N=\)\(65000000\). Here a and n denote the standard-form coefficient and exponent, not given constants.

Find n in that standard form.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle N=\)\(6600\). Here a and n denote the standard-form coefficient and exponent, not given constants.

Find n in that standard form.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle N=\)\(6600\). Here a and n denote the standard-form coefficient and exponent, not given constants.

Find 100N in ordinary notation.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle N=\)\(67000\). Here a and n denote the standard-form coefficient and exponent, not given constants.

Find 100N in ordinary notation.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle N=\)\(67000\). Here a and n denote the standard-form coefficient and exponent, not given constants.

Write \(\displaystyle \frac{1}{N}\) in standard form. Find its exponent.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle N=\)\(680000\). Here a and n denote the standard-form coefficient and exponent, not given constants.

Write \(\displaystyle \frac{1}{N}\) in standard form. Find its exponent.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Keep all values exact. Each square root denotes its nonnegative real value.

Write \(\displaystyle \sqrt{75}\) as \(\displaystyle k\sqrt{3}\). Find k.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Keep all values exact. Each square root denotes its nonnegative real value.

Write \(\displaystyle \sqrt{180}\) as \(\displaystyle k\sqrt{5}\). Find k.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Keep all values exact. Each square root denotes its nonnegative real value.

Find the coefficient of \(\displaystyle \sqrt{5}\) in \(\displaystyle 6\sqrt{5}+4\sqrt{5}\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Keep all values exact. Each square root denotes its nonnegative real value.

Find the coefficient of \(\displaystyle \sqrt{6}\) in \(\displaystyle 7\sqrt{6}+5\sqrt{6}\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Keep all values exact. Each square root denotes its nonnegative real value.

Find \(\displaystyle \sqrt{294} \times \sqrt{150}\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Keep all values exact. Each square root denotes its nonnegative real value.

Find \(\displaystyle \sqrt{128} \times \sqrt{72}\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Keep all values exact. Each square root denotes its nonnegative real value.

Find the coefficient of \(\displaystyle \sqrt{2}\) in \(\displaystyle 8\sqrt{2}-6\sqrt{2}\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Keep all values exact. Each square root denotes its nonnegative real value.

Find the coefficient of \(\displaystyle \sqrt{3}\) in \(\displaystyle 2\sqrt{3}-7\sqrt{3}\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle a=3, d=5\). Write \(\displaystyle \frac{1}{a+\sqrt d}=A+B\sqrt d\), where A and B are rational.

Find A in the stated rationalised form.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle a=4, d=6\). Write \(\displaystyle \frac{1}{a+\sqrt d}=A+B\sqrt d\), where A and B are rational.

Find A in the stated rationalised form.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle a=4, d=6\). Write \(\displaystyle \frac{1}{a+\sqrt d}=A+B\sqrt d\), where A and B are rational.

Find B in the stated rationalised form.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle a=5, d=2\). Write \(\displaystyle \frac{1}{a+\sqrt d}=A+B\sqrt d\), where A and B are rational.

Find B in the stated rationalised form.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle a=5, d=2\). Write \(\displaystyle \frac{1}{a+\sqrt d}=A+B\sqrt d\), where A and B are rational.

Find the rational product \(\displaystyle (a+\sqrt{d})(a-\sqrt{d})\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle a=6, d=3\). Write \(\displaystyle \frac{1}{a+\sqrt d}=A+B\sqrt d\), where A and B are rational.

Find the rational product \(\displaystyle (a+\sqrt{d})(a-\sqrt{d})\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle a=6, d=3\). Write \(\displaystyle \frac{1}{a+\sqrt d}=A+B\sqrt d\), where A and B are rational.

Write \(\displaystyle \frac{2}{\sqrt{d}}\) as \(\displaystyle K\sqrt{d}\). Find K.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

Let \(\displaystyle a=2, d=5\). Write \(\displaystyle \frac{1}{a+\sqrt d}=A+B\sqrt d\), where A and B are rational.

Write \(\displaystyle \frac{3}{\sqrt{d}}\) as \(\displaystyle K\sqrt{d}\). Find K.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

For nonzero x and y, F is the product of \(4 x^{3} y^{4}\) and \(\displaystyle \frac{5 x^{3}}{y^{4}}\), divided by \(2 x^{4}\). Write its simplified form as \(Kx^py^q\).

Find K in the simplified form Kxᵖyᑫ.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

For nonzero x and y, F is the product of \(5 x^{4} y^{5}\) and \(\displaystyle \frac{2 x^{4}}{y}\), divided by \(3 x\). Write its simplified form as \(Kx^py^q\).

Find K in the simplified form Kxᵖyᑫ.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

For nonzero x and y, F is the product of \(5 x^{4} y^{5}\) and \(\displaystyle \frac{2 x^{4}}{y}\), divided by \(3 x\). Write its simplified form as \(Kx^py^q\).

Find p in that form.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

For nonzero x and y, F is the product of \(6 x^{2} y^{6}\) and \(\displaystyle \frac{3 x^{5}}{y^{2}}\), divided by \(x^{2}\). Write its simplified form as \(Kx^py^q\).

Find p in that form.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

For nonzero x and y, F is the product of \(6 x^{2} y^{6}\) and \(\displaystyle \frac{3 x^{5}}{y^{2}}\), divided by \(x^{2}\). Write its simplified form as \(Kx^py^q\).

Find q in that form.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

For nonzero x and y, F is the product of \(2 x^{3} y^{2}\) and \(\displaystyle \frac{4 x}{y^{3}}\), divided by \(2 x^{3}\). Write its simplified form as \(Kx^py^q\).

Find q in that form.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

For nonzero x and y, F is the product of \(2 x^{3} y^{2}\) and \(\displaystyle \frac{4 x}{y^{3}}\), divided by \(2 x^{3}\). Write its simplified form as \(Kx^py^q\).

Find F when \(\displaystyle x=2\) and \(\displaystyle y=3\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

For nonzero x and y, F is the product of \(3 x^{4} y^{3}\) and \(\displaystyle \frac{5 x^{2}}{y^{4}}\), divided by \(3 x^{4}\). Write its simplified form as \(Kx^py^q\).

Find F when \(\displaystyle x=2\) and \(\displaystyle y=3\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

For x>0, multiply \(2 x\) by \(\sqrt[6]{x}\), then divide by \(2 \sqrt[3]{x}\). Call the result F and write it as \(Kx^q\).

Find K in the simplified form F=Kxᑫ.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

For x>0, multiply \(5 \sqrt{x}\) by \(\sqrt[6]{x}\), then divide by \(\displaystyle 2 x^{\frac{4}{3}}\). Call the result F and write it as \(Kx^q\).

Find K in the simplified form F=Kxᑫ.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

For x>0, multiply \(\displaystyle 3 x^{\frac{3}{2}}\) by \(\sqrt[3]{x}\), then divide by \(\displaystyle 3 x^{\frac{2}{3}}\). Call the result F and write it as \(Kx^q\).

Find q in that form.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

For x>0, multiply \(4 x^{2}\) by \(\sqrt{x}\), then divide by \(4 x\). Call the result F and write it as \(Kx^q\).

Find q in that form.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

For x>0, multiply \(4 x^{2}\) by \(\sqrt{x}\), then divide by \(4 x\). Call the result F and write it as \(Kx^q\).

Find the exponent of x when \(\displaystyle F^{2}\) is simplified.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

For x>0, multiply \(5 \sqrt{x}\) by \(\sqrt[6]{x}\), then divide by \(\displaystyle 2 x^{\frac{4}{3}}\). Call the result F and write it as \(Kx^q\).

Find the exponent of x when \(\displaystyle F^{2}\) is simplified.

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

For x>0, multiply \(5 \sqrt{x}\) by \(\sqrt[6]{x}\), then divide by \(\displaystyle 2 x^{\frac{4}{3}}\). Call the result F and write it as \(Kx^q\).

Find F exactly at \(\displaystyle x=64\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.

Maths4U Original · Short practice

For x>0, multiply \(6 x\) by \(\sqrt[3]{x}\), then divide by \(\displaystyle 3 x^{\frac{5}{3}}\). Call the result F and write it as \(Kx^q\).

Find F exactly at \(\displaystyle x=64\).

Enter one exact number or fraction unless the question specifies rounding. If it asks for a coefficient of π, enter that coefficient only. Give full reasoning separately for written problems.