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Maths 9 โ€“ Calculation exercises

๐Ÿ“˜ Notes

Evaluating Expressions with Square Roots and Powers

In these questions, we simplify expressions using square roots, decimal numbers, and powers. The main idea is to simplify roots first, then powers, and then complete the arithmetic carefully.

Always work step by step and follow the order of operations.

Key rules

\[ \sqrt{a}\cdot \sqrt{b}=\sqrt{ab} \] \[ \frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}} \] \[ a^m \div a^n=a^{m-n} \]
  • Multiply or divide square roots first when possible.
  • When dividing powers with the same base, subtract the indices.
  • Be careful with decimal numbers under square roots.
  • At the end, complete the subtraction or addition.

Useful steps

  1. Rewrite the square roots using the root rules.
  2. Simplify powers with the same base using index laws.
  3. Change roots and powers into ordinary numbers.
  4. Finish with the remaining addition or subtraction.
  5. Check that the final answer is sensible.

Worked examples

Example 1

Evaluate \( \sqrt{2}\cdot \sqrt{8}-1.25 \).

\[ \sqrt{2}\cdot \sqrt{8}=\sqrt{16}=4 \] \[ 4-1.25=2.75 \]

Example 2

Evaluate \( \dfrac{\sqrt{200}}{\sqrt{2}}-0.2 \).

\[ \frac{\sqrt{200}}{\sqrt{2}}=\sqrt{\frac{200}{2}}=\sqrt{100}=10 \] \[ 10-0.2=9.8 \]

Example 3

Evaluate \( \sqrt{49}-0.5^5 \div 0.5^3 \).

\[ \sqrt{49}=7 \] \[ 0.5^5 \div 0.5^3=0.5^{5-3}=0.5^2=0.25 \] \[ 7-0.25=6.75 \]

Example 4

Evaluate \( \sqrt{500}\cdot \sqrt{\frac{5}{4}} \).

\[ \sqrt{500}\cdot \sqrt{\frac{5}{4}}=\sqrt{500\cdot \frac{5}{4}}=\sqrt{625}=25 \]

Common mistakes

  • Forgetting that \( \sqrt{a}\cdot \sqrt{b}=\sqrt{ab} \).
  • Dividing powers incorrectly. Remember to subtract the indices.
  • Making mistakes with decimals such as \(0.36\) and \(0.036\).
  • Doing subtraction too early before simplifying the roots and powers.