Math Problem Statement

Simplify the expression. Write the answer in exponential notation

Simplify the expression. Write the answer in exponential notation

Solution

The expression in the image is:

(35)(35)37(35)22\frac{\left( \frac{3}{5} \right) \cdot \left( \frac{3}{5} \right)^{3^7}}{\left( \frac{3}{5} \right)^{2^2}}

Let's go step-by-step to simplify this expression using the laws of exponents.

Step 1: Multiplying powers of the same base.

(35)(35)37=(35)1+37\left( \frac{3}{5} \right) \cdot \left( \frac{3}{5} \right)^{3^7} = \left( \frac{3}{5} \right)^{1 + 3^7} Thus, the numerator becomes: (35)1+37\left( \frac{3}{5} \right)^{1 + 3^7}

Step 2: Simplifying the denominator.

(35)22=(35)4\left( \frac{3}{5} \right)^{2^2} = \left( \frac{3}{5} \right)^4

Step 3: Using the division rule for exponents.

(35)1+37(35)4=(35)(1+37)4\frac{\left( \frac{3}{5} \right)^{1 + 3^7}}{\left( \frac{3}{5} \right)^4} = \left( \frac{3}{5} \right)^{(1 + 3^7) - 4}

Step 4: Simplifying the exponent.

The exponent simplifies to 1+374=3731 + 3^7 - 4 = 3^7 - 3.

Thus, the final simplified expression in exponential notation is: (35)373\left( \frac{3}{5} \right)^{3^7 - 3}

Let me know if you need further clarification!

Here are some related questions:

  1. How do you simplify expressions with multiple powers?
  2. What are the rules of exponents for multiplying and dividing powers?
  3. What happens if the exponents in the numerator and denominator are equal?
  4. Can this expression be further simplified into a decimal form?
  5. How do exponent rules apply when dealing with fractional bases?

Tip: Always break down exponent operations using exponent rules before simplifying.

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Math Problem Analysis

Mathematical Concepts

Exponents
Fractional Bases
Exponent Rules

Formulas

Multiplication of exponents: a^m * a^n = a^(m+n)
Division of exponents: a^m / a^n = a^(m-n)

Theorems

Laws of exponents

Suitable Grade Level

Grades 9-12