Math Problem Statement

Simplify the expression completely: (y^3)^5 / (y^8)^7

Solution

The given expression is:

(y3)5(y8)7\frac{(y^3)^5}{(y^8)^7}

Let's simplify it step by step:

Step 1: Apply the Power of a Power Rule

The power of a power rule states (xm)n=xmn(x^m)^n = x^{m \cdot n}. So we apply that to both the numerator and the denominator:

(y3)5=y35=y15(y^3)^5 = y^{3 \cdot 5} = y^{15} (y8)7=y87=y56(y^8)^7 = y^{8 \cdot 7} = y^{56}

Thus, the expression becomes:

y15y56\frac{y^{15}}{y^{56}}

Step 2: Apply the Quotient of Powers Rule

The quotient of powers rule states xmxn=xmn\frac{x^m}{x^n} = x^{m-n}. Therefore:

y15y56=y1556=y41\frac{y^{15}}{y^{56}} = y^{15 - 56} = y^{-41}

Final Answer:

y41y^{-41}

This can also be written as:

1y41\frac{1}{y^{41}}

Would you like a detailed explanation of any step?

Here are five related questions to deepen your understanding:

  1. How do you apply the power of a power rule in different expressions?
  2. What does a negative exponent signify?
  3. How do you simplify expressions with different bases in the numerator and denominator?
  4. How can fractional exponents be handled similarly to integer exponents?
  5. How do exponent rules apply when both powers in the fraction are negative?

Tip: When simplifying exponent expressions, always apply the power rules carefully and reduce step by step to avoid errors.

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Math Problem Analysis

Mathematical Concepts

Exponentiation
Algebra
Simplifying Rational Expressions

Formulas

Power of a power rule: (x^m)^n = x^(m * n)
Quotient of powers rule: (x^m) / (x^n) = x^(m - n)

Theorems

Laws of Exponents

Suitable Grade Level

Grades 8-10