Can you use the different types of working only using the bracket

Question

ScanSolve AI Tutor · Accepted Answer

Okay Aisha, I understand you want to see different ways to work through this problem, specifically focusing on how the operations are applied within the brackets.

Let's simplify $\left(\frac{81}{16}\right)^{-\frac{3}{4}}$ using two common approaches for fractional exponents.

Method 1: Taking the root first, then the power

This method is generally preferred as it deals with smaller numbers.

Apply the negative exponent rule, $a^{-n} = \frac{1}{a^n}$ , to invert the fraction inside the bracket.
$\left(\frac{81}{16}\right)^{-\frac{3}{4}} = \left(\frac{16}{81}\right)^{\frac{3}{4}}$
Rewrite the fractional exponent $\frac{3}{4}$ as $\frac{1}{4} \times 3$ .
This means we take the fourth root first, then cube the result.
$\left(\frac{16}{81}\right)^{\frac{3}{4}} = \left(\left(\frac{16}{81}\right)^{\frac{1}{4}}\right)^3$
Calculate the fourth root of the fraction inside the inner bracket.
$\left(\left(\frac{16}{81}\right)^{\frac{1}{4}}\right)^3 = \left(\frac{\sqrt[4]{16}}{\sqrt[4]{81}}\right)^3 = \left(\frac{2}{3}\right)^3$
Cube the resulting fraction.
$\left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27}$

Method 2: Taking the power first, then the root

This method can involve larger numbers but is mathematically equivalent.
Apply the negative exponent rule, $a^{-n} = \frac{1}{a^n}$ , to invert the fraction inside the bracket.
$\left(\frac{81}{16}\right)^{-\frac{3}{4}} = \left(\frac{16}{81}\right)^{\frac{3}{4}}$
Rewrite the fractional exponent $\frac{3}{4}$ as $3 \times \frac{1}{4}$ .
This means we cube the fraction first, then take the fourth root of the result.
$\left(\frac{16}{81}\right)^{\frac{3}{4}} = \left(\left(\frac{16}{81}\right)^3\right)^{\frac{1}{4}}$
Cube the fraction inside the inner bracket.
$\left(\left(\frac{16}{81}\right)^3\right)^{\frac{1}{4}} = \left(\frac{16^3}{81^3}\right)^{\frac{1}{4}} = \left(\frac{4096}{531441}\right)^{\frac{1}{4}}$
Calculate the fourth root of the resulting fraction.
$\left(\frac{4096}{531441}\right)^{\frac{1}{4}} = \frac{\sqrt[4]{4096}}{\sqrt[4]{531441}} = \frac{8}{27}$

Both methods lead to the same simplified answer:
$\frac{8}{27}$

Can you use the different types of working only using the bracket

Answer

Step-by-step solution

Method 1: Taking the root first, then the power

Apply the negative exponent rule, $a^{-n} = \frac{1}{a^n}$ , to invert the fraction inside the bracket.

Rewrite the fractional exponent $\frac{3}{4}$ as $\frac{1}{4} \times 3$ .

Calculate the fourth root of the fraction inside the inner bracket.

Cube the resulting fraction.

Method 2: Taking the power first, then the root

Apply the negative exponent rule, $a^{-n} = \frac{1}{a^n}$ , to invert the fraction inside the bracket.

Rewrite the fractional exponent $\frac{3}{4}$ as $3 \times \frac{1}{4}$ .

Cube the fraction inside the inner bracket.

Calculate the fourth root of the resulting fraction.

Try one yourself

Still stuck on this one?

Can you use the different types of working only using the bracket

Answer

Step-by-step solution

Method 1: Taking the root first, then the power

Apply the negative exponent rule, a−n=1ana^{-n} = \frac{1}{a^n}a−n=an1​, to invert the fraction inside the bracket.

Rewrite the fractional exponent 34\frac{3}{4}43​ as 14×3\frac{1}{4} \times 341​×3.

Calculate the fourth root of the fraction inside the inner bracket.

Cube the resulting fraction.

Method 2: Taking the power first, then the root

Apply the negative exponent rule, a−n=1ana^{-n} = \frac{1}{a^n}a−n=an1​, to invert the fraction inside the bracket.

Rewrite the fractional exponent 34\frac{3}{4}43​ as 3×143 \times \frac{1}{4}3×41​.

Cube the fraction inside the inner bracket.

Calculate the fourth root of the resulting fraction.

Try one yourself

Still stuck on this one?

Apply the negative exponent rule, $a^{-n} = \frac{1}{a^n}$ , to invert the fraction inside the bracket.

Rewrite the fractional exponent $\frac{3}{4}$ as $\frac{1}{4} \times 3$ .

Apply the negative exponent rule, $a^{-n} = \frac{1}{a^n}$ , to invert the fraction inside the bracket.

Rewrite the fractional exponent $\frac{3}{4}$ as $3 \times \frac{1}{4}$ .