Math Problem Statement

how to do 2x to the power of 2/3 to the power of -3

Solution

The expression you've given is (2x23)3(2x^{\frac{2}{3}})^{-3}. To simplify this, we can break it down step by step:

Step 1: Apply the power of a power rule

The power of a power rule states that when you raise a power to another power, you multiply the exponents. Here's how it works:

(2x23)3=23(x23)3(2x^{\frac{2}{3}})^{-3} = 2^{-3} \cdot (x^{\frac{2}{3}})^{-3}

Step 2: Simplify each term

Now simplify each part:

  • For 232^{-3}, calculate the power of 2: 23=123=182^{-3} = \frac{1}{2^3} = \frac{1}{8}

  • For (x23)3(x^{\frac{2}{3}})^{-3}, multiply the exponents 23×(3)\frac{2}{3} \times (-3): x23×3=x2x^{\frac{2}{3} \times -3} = x^{-2}

Step 3: Combine the results

Now combine the results: 18x2=18x2\frac{1}{8} \cdot x^{-2} = \frac{1}{8x^2}

Final Answer:

The simplified form of (2x23)3(2x^{\frac{2}{3}})^{-3} is: 18x2\frac{1}{8x^2}

Would you like a more detailed explanation or have any questions?

Here are 5 related questions to practice:

  1. Simplify (3x12)4(3x^{\frac{1}{2}})^{-4}.
  2. What is the value of (5x34)2(5x^{\frac{3}{4}})^{-2}?
  3. Simplify (4x56)3(4x^{\frac{5}{6}})^{-3}.
  4. How do you simplify (2x1)3(2x^{-1})^{-3}?
  5. Simplify (6x25)52(6x^{\frac{2}{5}})^{\frac{5}{2}}.

Tip: When raising a term to a negative power, you can flip the term and change the exponent to positive!

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Math Problem Analysis

Mathematical Concepts

Exponents
Power Rules
Negative Exponents

Formulas

Power of a power rule: (a^m)^n = a^(m*n)
Negative exponent rule: a^(-n) = 1/(a^n)

Theorems

Exponentiation Theorems

Suitable Grade Level

Grades 8-10