Math Problem Statement

Simplify the expression \(\frac{(8x^3y^2)^2}{(2x)^5y}\)

Solution

To solve the given problem, we need to simplify the expression:

(8x3y2)2(2x)5y\frac{(8x^3y^2)^2}{(2x)^5y}

Let's go through it step-by-step.

Step 1: Simplify the Numerator

The numerator is (8x3y2)2(8x^3y^2)^2. Apply the power of a product rule (ab)n=anbn(ab)^n = a^n b^n, which gives:

(8x3y2)2=82(x3)2(y2)2(8x^3y^2)^2 = 8^2 \cdot (x^3)^2 \cdot (y^2)^2

Calculate each term:

  • 82=648^2 = 64
  • (x3)2=x3×2=x6(x^3)^2 = x^{3 \times 2} = x^6
  • (y2)2=y2×2=y4(y^2)^2 = y^{2 \times 2} = y^4

So, the numerator becomes:

64x6y464x^6y^4

Step 2: Simplify the Denominator

The denominator is (2x)5y(2x)^5 y. Again, apply the power of a product rule:

(2x)5=25x5(2x)^5 = 2^5 \cdot x^5

Calculate each term:

  • 25=322^5 = 32
  • x5x^5 remains as it is.

So, the denominator becomes:

32x5y32x^5y

Step 3: Divide the Numerator by the Denominator

Now we have:

64x6y432x5y\frac{64x^6y^4}{32x^5y}

Divide each term separately:

  • Constants: 6432=2\frac{64}{32} = 2
  • xx-terms: x6x5=x65=x\frac{x^6}{x^5} = x^{6-5} = x
  • yy-terms: y4y=y41=y3\frac{y^4}{y} = y^{4-1} = y^3

Putting it all together, the expression simplifies to:

2xy32xy^3

Final Answer

2xy3\boxed{2xy^3}

Would you like further details or have any questions?

Here are some related questions for further practice:

  1. How would you simplify (3x4y)3(x2y2)2\frac{(3x^4y)^3}{(x^2y^2)^2}?
  2. What is the result of (4x2y3)3/(2x5y)(4x^2y^3)^3 / (2x^5y)?
  3. How do you simplify (10a2b)3(5a)4b2\frac{(10a^2b)^3}{(5a)^4b^2}?
  4. Simplify (7x2y)2(x3y)3\frac{(7x^2y)^2}{(x^3y)^3}.
  5. What steps would you take to simplify (5x4y3)2(x6y)\frac{(5x^4y^3)^2}{(x^6y)}?

Tip: When simplifying expressions with exponents, remember to carefully apply the power of a product and the division of powers rules to avoid common mistakes.

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Math Problem Analysis

Mathematical Concepts

Exponentiation
Algebraic Simplification
Division of Exponents

Formulas

Power of a Product Rule: \((ab)^n = a^n b^n\)
Power Rule for Exponents: \((x^m)^n = x^{m \times n}\)
Division of Exponents Rule: \(\frac{x^m}{x^n} = x^{m - n}\)

Theorems

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Suitable Grade Level

Grades 9-11