Math Problem Statement

Use exponent properties to simplify and evaluate the following expression: 5^4 × (5^5 ÷ 5^7)^2 ÷ 5^5.

Solution

Let's solve the given expression step by step:

Problem:

54×(5557)2÷555^4 \times \left( \frac{5^5}{5^7} \right)^2 \div 5^5

Step 1: Simplify 5557\frac{5^5}{5^7}

Using the property of exponents: aman=amn\frac{a^m}{a^n} = a^{m-n}, 5557=557=52.\frac{5^5}{5^7} = 5^{5-7} = 5^{-2}.

Step 2: Square the result

Now square 525^{-2}: (52)2=522=54.\left( 5^{-2} \right)^2 = 5^{-2 \cdot 2} = 5^{-4}.

Step 3: Multiply 54×545^4 \times 5^{-4}

Using the property of exponents: am×an=am+na^m \times a^n = a^{m+n}, 54×54=544=50.5^4 \times 5^{-4} = 5^{4-4} = 5^0.

Note that 50=15^0 = 1.

Step 4: Divide by 555^5

Now divide 1÷551 \div 5^5: 155.\frac{1}{5^5}.

Final Answer:

155.\frac{1}{5^5}.

Boxed Answer:

Numerator: 11, Denominator: 555^5.


Do you want a breakdown of these steps, or further clarification? 😊

Related Questions:

  1. What are the basic exponent rules and how are they applied in simplifying expressions?
  2. Why does a0=1a^0 = 1 for any nonzero value aa?
  3. How does squaring a term with a negative exponent affect the final result?
  4. Can this problem be solved without using exponent rules? How would that look?
  5. How does dividing by a power compare to multiplying by a negative power?

Tip:

When dealing with multiple operations on exponents, always work systematically: simplify powers, apply multiplication/division, and only then deal with fractions or squaring.

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Math Problem Analysis

Mathematical Concepts

Exponent Properties
Simplification of Expressions
Division of Exponents
Negative Exponents

Formulas

a^m ÷ a^n = a^(m-n)
(a^m)^n = a^(m⋅n)
a^m × a^n = a^(m+n)
a^0 = 1

Theorems

Exponent Laws

Suitable Grade Level

Grades 8-10