Math Problem Statement

Simplify each of the following expressions: (5x)^2, (4y)^3, (7ab)^2, (-2x)^3, (-2x)^4, (10ab)^4, (6xyz)^2, (-xy)^5, (7rs)^3, (5x)^{-2}

Solution

Let's solve each of these expressions step-by-step. These involve using exponent rules for products, such as:

(ab)n=anbn(a \cdot b)^n = a^n \cdot b^n


1. (5x)2(5x)^2

(5x)2=52x2=25x2(5x)^2 = 5^2 \cdot x^2 = 25x^2

2. (4y)3(4y)^3

(4y)3=43y3=64y3(4y)^3 = 4^3 \cdot y^3 = 64y^3

3. (7ab)2(7ab)^2

(7ab)2=72a2b2=49a2b2(7ab)^2 = 7^2 \cdot a^2 \cdot b^2 = 49a^2b^2

4. (2x)3(-2x)^3

(2x)3=(2)3x3=8x3(-2x)^3 = (-2)^3 \cdot x^3 = -8x^3

5. (2x)4(-2x)^4

(2x)4=(2)4x4=16x4(-2x)^4 = (-2)^4 \cdot x^4 = 16x^4

6. (10ab)4(10ab)^4

(10ab)4=104a4b4=10000a4b4(10ab)^4 = 10^4 \cdot a^4 \cdot b^4 = 10000a^4b^4

7. (6xyz)2(6xyz)^2

(6xyz)2=62x2y2z2=36x2y2z2(6xyz)^2 = 6^2 \cdot x^2 \cdot y^2 \cdot z^2 = 36x^2y^2z^2

8. (xy)5(-xy)^5

(xy)5=(1)5x5y5=x5y5(-xy)^5 = (-1)^5 \cdot x^5 \cdot y^5 = -x^5y^5

9. (7rs)3(7rs)^3

(7rs)3=73r3s3=343r3s3(7rs)^3 = 7^3 \cdot r^3 \cdot s^3 = 343r^3s^3

10. (5x)2(5x)^{-2}

(5x)2=1(5x)2=125x2(5x)^{-2} = \frac{1}{(5x)^2} = \frac{1}{25x^2}


If you need further clarifications or additional explanations, feel free to ask!


Related Questions:

  1. What happens when an exponent is zero?
  2. How do negative exponents affect the expression?
  3. What is the rule for raising a product to a power?
  4. How do exponents apply to fractional bases?
  5. What happens if an exponent is negative and the base includes a negative number?

Tip: When raising a product to a power, apply the exponent to each factor individually to simplify the expression.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Exponents
Algebra
Simplifying Expressions

Formulas

(a * b)^n = a^n * b^n
Negative exponent rule: a^{-n} = 1 / a^n
Power of a product rule: (ab)^n = a^n * b^n

Theorems

Exponent Rules

Suitable Grade Level

Grades 7-9