Math Problem Statement

(x^2 y z^3)^5

Solution

The given expression is (x2yz3)5(x^2 y z^3)^5. To simplify it, we can apply the power of a product rule, which states:

(abc)n=anbncn(a \cdot b \cdot c)^n = a^n \cdot b^n \cdot c^n

We'll distribute the exponent 55 to each part inside the parentheses:

  1. For x2x^2, we raise x2x^2 to the power of 5: (x2)5=x25=x10(x^2)^5 = x^{2 \cdot 5} = x^{10}
  2. For yy, we raise yy to the power of 5: y15=y5y^1 \cdot 5 = y^5
  3. For z3z^3, we raise z3z^3 to the power of 5: (z3)5=z35=z15(z^3)^5 = z^{3 \cdot 5} = z^{15}

So the simplified expression is: x10y5z15x^{10} y^5 z^{15}

It appears that in the image, the exponent for yy was mistakenly written as 2. The correct final answer should be:

x10y5z15x^{10} y^5 z^{15}

Would you like further clarification on this?

Here are 5 follow-up questions based on this explanation:

  1. What is the power rule for exponents?
  2. How do you simplify expressions involving exponents inside parentheses?
  3. Can the exponent 55 apply to any other operations besides multiplication?
  4. How would the result change if the exponent applied to a sum instead of a product?
  5. What is the general rule for distributing exponents over multiple variables?

Tip: Always double-check the exponents for all variables when simplifying expressions to avoid small mistakes!

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Math Problem Analysis

Mathematical Concepts

Exponentiation
Power Rule

Formulas

(a^m)^n = a^{m*n}
(ab)^n = a^n * b^n

Theorems

Power of a Product Rule
Exponentiation Theorem

Suitable Grade Level

Grades 8-10