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