Question

Which of these choices show a pair if equivelent expressions? Chack all that apply

Answer 1

Answer:

A. $(\sqrt[3]{125})^9\ and\ (125)^{\frac{9}{3}}$

D. $8^{\frac{9}{2}}\ and\ (\sqrt{8})^9$

Step-by-step explanation:

Equivalent expressions are those expressions that simplify to same form.

Now, let us check each of the given options.

Option A:

$(\sqrt[3]{125})^9\ and\ (125)^{\frac{9}{3}}$

We know that,

$\sqrt[n]{x} =x^{\frac{1}{n}}$

Therefore, $\sqrt[3]{125} =(125)^{\frac{1}{3}}$

Thus the first expression becomes;

$((125)^{\frac{1}{3}})^9$

Now, using law of indices $(a^m)^n=a^{m\times n}$, we get

$((125)^{\frac{1}{3}})^9=((125))^{\frac{1}{3}\times 9}=((125))^{\frac{9}{3}$

Therefore, $(\sqrt[3]{125})^9\ and\ (125)^{\frac{9}{3}}$ are equivalent.

Option B:

$12^{\frac{2}{7}}\ and\ (\sqrt{12})^7$

Consider the second expression $(\sqrt{12})^7$

We know that,

$\sqrt x=x^{\frac{1}{2}}$

$(\sqrt{12})^7=((12)^{\frac{1}{2}})^7=(12)^{\frac{1}{2}\times 7}=(12)^{\frac{7}{2}}$

Therefore, $12^{\frac{2}{7}} \ne (12)^{\frac{7}{2}}$. Hence, the expressions $12^{\frac{2}{7}}\ and\ (\sqrt{12})^7$ are not equivalent.

Option C:

$4^{\frac{1}{5}}\ and\ (\sqrt 4)^5$

We know that,

$x^{\frac{1}{n}}=\sqrt[n]{x}$

Therefore, $4^{\frac{1}{5}}=\sqrt[5]{4}$

Now, $\sqrt[5]{4}\ne (\sqrt 4)^5$

Therefore, the expressions $4^{\frac{1}{5}}\ and\ (\sqrt 4)^5$ are not equivalent.

Option D:

$8^{\frac{9}{2} }\ and\ (\sqrt 8)^9$

Using law of indices $a^{m\times n}=(a^m)^n$, we get

$8^{\frac{9}{2} }=(8^{\frac{1}{2}})^9$

Now, we know that, $x^{\frac{1}{2}}=\sqrt x$

So, $(8^{\frac{1}{2}})^9=(\sqrt8)^9$

Therefore, $8^{\frac{9}{2} }\ and\ (\sqrt 8)^9$ are equivalent.