Math Problem Statement

solve

Simplify the following expression using the index laws: \( ( \sqrt{5} \times \sqrt[3]{5} )^5 \)

Solution

Let's solve the given expression step by step. The problem is:

(5×53)5\left( \sqrt{5} \times \sqrt[3]{5} \right)^5

Step 1: Rewrite the terms in fractional powers

  • 5\sqrt{5} can be rewritten as 51/25^{1/2}.
  • 53\sqrt[3]{5} can be rewritten as 51/35^{1/3}.

Thus, the expression becomes: (51/2×51/3)5\left( 5^{1/2} \times 5^{1/3} \right)^5

Step 2: Apply the index law for multiplication

When multiplying terms with the same base, you add their exponents: 51/2×51/3=5(1/2+1/3)5^{1/2} \times 5^{1/3} = 5^{(1/2 + 1/3)}

To simplify 1/2+1/31/2 + 1/3, find the common denominator: 1/2+1/3=3/6+2/6=5/61/2 + 1/3 = 3/6 + 2/6 = 5/6

So, the expression becomes: (55/6)5\left( 5^{5/6} \right)^5

Step 3: Apply the index law for powers

When raising a power to another power, multiply the exponents: (55/6)5=5(5/65)\left( 5^{5/6} \right)^5 = 5^{(5/6 \cdot 5)}

Simplify 5/655/6 \cdot 5: 5/65=25/65/6 \cdot 5 = 25/6

Thus, the expression simplifies to: 525/65^{25/6}

Final Answer:

525/65^{25/6}


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Related Questions:

  1. What are the laws of indices used in this solution?
  2. How do you simplify fractional exponents step by step?
  3. What happens if the exponents involve negative powers?
  4. Can this expression 525/65^{25/6} be written in radical form?
  5. What is the numerical value of 525/65^{25/6} to 3 decimal places?

Tip:

When simplifying expressions with powers and roots, always rewrite terms in their fractional exponent form to make the calculations easier.

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Math Problem Analysis

Mathematical Concepts

Fractional Exponents
Index Laws
Simplifying Expressions

Formulas

a^{m} \times a^{n} = a^{m+n}
(a^{m})^{n} = a^{m \cdot n}

Theorems

Laws of Indices

Suitable Grade Level

Grade 9-11