Question

what is the value of the expression $-sqrt3{125^{5}}$?
$-sqrt3{125^{5}}=square$
(type an integer or a decimal.)

Explanation:

Step1: Recognize the cube root of 125

We know that \(125 = 5^3\), so we can rewrite the expression inside the cube root.
\(-\sqrt[3]{125^{5}}=-\sqrt[3]{(5^{3})^{5}}\)

Step2: Apply the power of a power rule

Using the rule \((a^{m})^{n}=a^{mn}\), we simplify the exponent inside the cube root.
\(-\sqrt[3]{(5^{3})^{5}}=-\sqrt[3]{5^{15}}\)

Step3: Simplify the cube root

Since the cube root of \(a^{3n}\) is \(a^{n}\) (because \(\sqrt[3]{a^{3n}}=(a^{3n})^{\frac{1}{3}} = a^{n}\)), for \(a = 5\) and \(3n=15\) (so \(n = 5\)):
\(-\sqrt[3]{5^{15}}=-5^{5}\)

Step4: Calculate \(5^{5}\)

\(5^{5}=5\times5\times5\times5\times5 = 3125\)
So, \(-5^{5}=- 3125\)

Answer:

\(-3125\)