Math Problem Statement

Generate a numerical expression equivalent to (5^2 / 3^3)^3.

Solution

The expression given is:

(5233)3\left(\frac{5^2}{3^3}\right)^3

To simplify this:

  1. Start by simplifying the inner part: 52=25and33=275^2 = 25 \quad \text{and} \quad 3^3 = 27 So the fraction becomes: 2527\frac{25}{27}

  2. Raise this fraction to the power of 3: (2527)3=253273\left(\frac{25}{27}\right)^3 = \frac{25^3}{27^3}

  3. Calculate 25325^3 and 27327^3:

    • 253=25×25×25=1562525^3 = 25 \times 25 \times 25 = 15625
    • 273=27×27×27=1968327^3 = 27 \times 27 \times 27 = 19683

Thus, the simplified numerical expression is:

1562519683\frac{15625}{19683}

Let me know if you'd like further explanations or if you have more questions!

Here are five related questions to explore:

  1. How do you simplify expressions with exponents in the numerator and denominator?
  2. What are the properties of exponents used in similar problems?
  3. How do you evaluate higher powers like 25325^3 or 27327^3 efficiently?
  4. Can you represent 1562519683\frac{15625}{19683} in decimal form for comparison?
  5. How would this process change if the base numbers (5 and 3) were replaced with variables?

Tip: When simplifying exponentiation, always resolve the inner terms before applying the outer exponent to avoid unnecessary complexity.

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Math Problem Analysis

Mathematical Concepts

Exponents
Fraction simplification
Power of a fraction

Formulas

(a/b)^n = a^n / b^n
a^m * a^n = a^(m+n)

Theorems

Laws of exponents

Suitable Grade Level

Grades 9-10