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Work out the exact value of $x$ - Edexcel - GCSE Maths - Question 18 - 2017 - Paper 2

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Question 18

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Work out the exact value of $x$. $$16^{ rac{1}{5}} \cdot x^2 = 8^{\frac{2}{3}}$$

Worked Solution & Example Answer:Work out the exact value of $x$ - Edexcel - GCSE Maths - Question 18 - 2017 - Paper 2

Step 1

Convert to a common base

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Answer

To solve the equation, we'll first rewrite both sides using powers of 2. We know that:

  • 16=2416 = 2^4, thus 1615=(24)15=24516^{\frac{1}{5}} = (2^4)^{\frac{1}{5}} = 2^{\frac{4}{5}}
  • 8=238 = 2^3, thus 823=(23)23=228^{\frac{2}{3}} = (2^3)^{\frac{2}{3}} = 2^2

With these conversions, the equation can be rewritten as:

245⋅x2=222^{\frac{4}{5}} \cdot x^2 = 2^2

Step 2

Equate the exponents

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Answer

Since the bases are the same, we can set the exponents equal to each other:

45+2⋅log⁡2(x)=2\frac{4}{5} + 2 \cdot \log_2(x) = 2

Solving for log⁡2(x)\log_2(x) gives:

log⁡2(x)=1−45=15\log_2(x) = 1 - \frac{4}{5} = \frac{1}{5}

Step 3

Find the value of $x$

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Answer

To find xx, we convert back from logarithmic form:

x=215x = 2^{\frac{1}{5}}

Thus, the exact value of xx is:

x=25x = \sqrt[5]{2}

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