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2. (a) Evaluate $81^{ rac{3}{2}}$ (b) Simplify fully $x^2 \left(4x^{-\frac{1}{2}}\right)^2$ - Edexcel - A-Level Maths Pure - Question 4 - 2014 - Paper 2

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2.-(a)-Evaluate-$81^{-rac{3}{2}}$--(b)-Simplify-fully-$x^2-\left(4x^{-\frac{1}{2}}\right)^2$-Edexcel-A-Level Maths Pure-Question 4-2014-Paper 2.png

2. (a) Evaluate $81^{ rac{3}{2}}$ (b) Simplify fully $x^2 \left(4x^{-\frac{1}{2}}\right)^2$

Worked Solution & Example Answer:2. (a) Evaluate $81^{ rac{3}{2}}$ (b) Simplify fully $x^2 \left(4x^{-\frac{1}{2}}\right)^2$ - Edexcel - A-Level Maths Pure - Question 4 - 2014 - Paper 2

Step 1

Evaluate $81^{\frac{3}{2}}$

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Answer

To evaluate 813281^{\frac{3}{2}}, we can rewrite 8181 as 929^2. Thus, we have:

8132=(92)32=92⋅32=9381^{\frac{3}{2}} = (9^2)^{\frac{3}{2}} = 9^{2 \cdot \frac{3}{2}} = 9^3

Calculating 939^3 gives:

93=7299^3 = 729

Therefore, the answer is 729729.

Step 2

Simplify fully $x^2 \left(4x^{-\frac{1}{2}}\right)^2$

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Answer

First, we simplify the expression inside the parentheses:

(4x−12)2=42⋅(x−12)2=16⋅x−1\left(4x^{-\frac{1}{2}}\right)^2 = 4^2 \cdot (x^{-\frac{1}{2}})^2 = 16 \cdot x^{-1}

Now, substituting this back into the equation:

x2⋅16⋅x−1x^2 \cdot 16 \cdot x^{-1}

Next, we combine the terms:

16⋅x2−1=16x16 \cdot x^{2 - 1} = 16x

Thus, the final simplified expression is 16x16x.

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