Math Problem Statement

Do the odd numbers only

Solution

Sure! I'll solve the odd-numbered problems from the image. I'll start with problems that I can clearly see. Let's get started:

Problem 27:

Simplify:
(223362)3\left( \frac{2^2 \cdot 3^3}{6^2} \right)^3

Solution: First, let's simplify the expression inside the parentheses.

  • 2233=427=1082^2 \cdot 3^3 = 4 \cdot 27 = 108
  • 62=366^2 = 36

So the expression becomes: (10836)3\left( \frac{108}{36} \right)^3 Now, simplify 10836\frac{108}{36}: 10836=3\frac{108}{36} = 3 Therefore, the expression simplifies to: 33=273^3 = 27 Answer: 2727


Problem 29:

Simplify:
(2x2)24x3\frac{\left( -2x^2 \right)^2}{4x^3}

Solution: Expand the numerator: (2x2)2=(2)2(x2)2=4x4\left( -2x^2 \right)^2 = (-2)^2 \cdot (x^2)^2 = 4x^4 So the expression becomes: 4x44x3\frac{4x^4}{4x^3} Simplify by dividing both the coefficient and the exponent of xx: =x43=x1=x= x^{4-3} = x^1 = x

Answer: xx


Problem 31:

Simplify:
6a32b2\frac{6a^{-3}}{2b^{-2}}

Solution: Simplify the coefficients: 62=3\frac{6}{2} = 3 Now handle the exponents: a3=1a3,b2=1b2a^{-3} = \frac{1}{a^3}, \quad b^{-2} = \frac{1}{b^2} So the expression simplifies to: 31a3b2=3b2a33 \cdot \frac{1}{a^3} \cdot b^2 = \frac{3b^2}{a^3}

Answer: 3b2a3\frac{3b^2}{a^3}


Problem 33:

Simplify:
(a2b3a3b2)2\left( \frac{a^{-2} b^3}{a^{-3} b^2} \right)^2

Solution: First, simplify the expression inside the parentheses: a2b3a3b2=a2(3)b32=a1b1=ab\frac{a^{-2} b^3}{a^{-3} b^2} = a^{-2 - (-3)} \cdot b^{3-2} = a^{1} \cdot b^{1} = ab Now, square the result: (ab)2=a2b2(ab)^2 = a^2 b^2

Answer: a2b2a^2b^2


Problem 35:

Simplify: x2y4z3zy3\frac{x^{-2}y^4}{z^{-3}} \cdot \frac{z}{y^3}

Solution: First, simplify the expression: x2y4z3zy3=x2y4zz3y3\frac{x^{-2}y^4}{z^{-3}} \cdot \frac{z}{y^3} = \frac{x^{-2} y^4 \cdot z}{z^{-3} \cdot y^3} This simplifies to: =x2y43z1+31=x2y1z41=yz4x2= \frac{x^{-2} y^{4-3} \cdot z^{1+3}}{1} = \frac{x^{-2} y^1 z^4}{1} = \frac{y z^4}{x^2}

Answer: yz4x2\frac{yz^4}{x^2}


Problem 37:

Simplify:
(x1y)2y3\frac{(x^{-1} y)^2}{y^{-3}}

Solution: First, expand the numerator: (x1y)2=x2y2(x^{-1} y)^2 = x^{-2} y^2 So the expression becomes: x2y2y3=x2y2(3)=x2y5\frac{x^{-2} y^2}{y^{-3}} = x^{-2} y^{2 - (-3)} = x^{-2} y^{5}

Answer: y5x2\frac{y^5}{x^2}


These are the simplified results for the odd-numbered problems visible from the image.

Would you like more details on any of these steps, or do you have any further questions?

Here are 5 related questions to deepen your understanding:

  1. How do you simplify expressions with both negative and fractional exponents?
  2. Can you solve a similar expression involving fractional exponents?
  3. What happens when you multiply two expressions with different bases but the same exponents?
  4. How do you simplify a complex fraction involving exponents?
  5. Can you apply these exponent rules to solve equations rather than just simplifying?

Tip: When simplifying expressions involving exponents, always pay careful attention to the properties of exponents, such as the power of a power rule and the product of powers rule, to ensure accuracy.

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Math Problem Analysis

Mathematical Concepts

Exponents
Fractional Exponents
Simplifying Algebraic Expressions

Formulas

Exponent Rules
Fractional Exponent Rules

Theorems

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Suitable Grade Level

High School