Questions: Below are three mathematical expressions. Are these expressions equivalent? Jus (hint: write them all as a single base) Expression 3 ∛625^12 × √5^176

Solution Steps

Expression 1: \(\frac{5^{-10} \times 5^{112}}{25^{-1}}\)

Rewrite \(25^{-1}\) as \((5^2)^{-1} = 5^{-2}\):

\[ \frac{5^{-10} \times 5^{112}}{5^{-2}} \]

Combine the exponents in the numerator:

\[ 5^{-10 + 112} = 5^{102} \]

Now, divide by \(5^{-2}\):

\[ 5^{102} \div 5^{-2} = 5^{102 - (-2)} = 5^{102 + 2} = 5^{104} \]

Expression 2: \(125 \times 625 \div 5^{-97}\)

Rewrite \(125\) and \(625\) as powers of 5:

\[ 125 = 5^3 \quad \text{and} \quad 625 = 5^4 \]

So, the expression becomes:

\[ 5^3 \times 5^4 \div 5^{-97} \]

Combine the exponents in the numerator:

\[ 5^{3 + 4} = 5^7 \]

Now, divide by \(5^{-97}\):

\[ 5^7 \div 5^{-97} = 5^{7 - (-97)} = 5^{7 + 97} = 5^{104} \]

Expression 3: \(\sqrt[3]{625^{12}} \times \sqrt{5^{176}}\)

Rewrite \(625\) as a power of 5:

\[ 625 = 5^4 \]

So, the expression becomes:

\[ \sqrt[3]{(5^4)^{12}} \times \sqrt{5^{176}} \]

Simplify the exponents:

\[ \sqrt[3]{5^{4 \times 12}} = \sqrt[3]{5^{48}} = 5^{48/3} = 5^{16} \]

And:

\[ \sqrt{5^{176}} = 5^{176/2} = 5^{88} \]

Now, multiply the results:

\[ 5^{16} \times 5^{88} = 5^{16 + 88} = 5^{104} \]

Final Answer