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Solve the equation: $$16^{\frac{1}{3}} \cdot x^2 = 8^{\frac{2}{3}}$$ Work out the exact value of x. - Edexcel - GCSE Maths - Question 18 - 2017 - Paper 2

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Solve-the-equation:--$$16^{\frac{1}{3}}-\cdot-x^2-=-8^{\frac{2}{3}}$$--Work-out-the-exact-value-of-x.-Edexcel-GCSE Maths-Question 18-2017-Paper 2.png

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

Worked Solution & Example Answer:Solve the equation: $$16^{\frac{1}{3}} \cdot x^2 = 8^{\frac{2}{3}}$$ 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 start, we need to express both sides of the equation in terms of a common base. The number 16 can be rewritten as 16=2416 = 2^4, and the number 8 can be rewritten as 8=238 = 2^3. Therefore, we can rewrite the equation as:

(24)13⋅x2=(23)23\left(2^4\right)^{\frac{1}{3}} \cdot x^2 = \left(2^3\right)^{\frac{2}{3}}

This simplifies to:

243⋅x2=222^{\frac{4}{3}} \cdot x^2 = 2^{2}

Step 2

Equate the powers of x

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Answer

Now, we can express the equation more clearly as:

x2=22243x^2 = \frac{2^{2}}{2^{\frac{4}{3}}}

Using the laws of exponents, this becomes:

x2=22−43=263−43=223x^2 = 2^{2 - \frac{4}{3}} = 2^{\frac{6}{3} - \frac{4}{3}} = 2^{\frac{2}{3}}

Step 3

Solve for x

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Answer

Taking the square root of both sides to solve for x gives:

x=223=213x = \sqrt{2^{\frac{2}{3}}} = 2^{\frac{1}{3}}

Thus, the exact value of x is:

x=213x = 2^{\frac{1}{3}}

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