Math Problem Statement

work the problems out and solve

Work the problems out and solve the algebraic expressions.

Solution

Let's go through each problem from your worksheet and simplify them step-by-step.

1) 3xy+4xy+5x2y+6xy23xy + 4xy + 5x^2y + 6xy^2

Combine like terms:

  • 3xy+4xy=7xy3xy + 4xy = 7xy

The expression becomes: 7xy+5x2y+6xy27xy + 5x^2y + 6xy^2 This is simplified as much as possible.

2) 6(x2+y2)7(x2+y2)6(x^2 + y^2) - 7(x^2 + y^2)

Factor out the common term (x2+y2)(x^2 + y^2): (67)(x2+y2)=1(x2+y2)(6 - 7)(x^2 + y^2) = -1(x^2 + y^2) The simplified expression is: (x2+y2)-(x^2 + y^2)

3) 18x2y224xy2\frac{18x^2y^2}{24xy^2}

Simplify by canceling out common terms:

  • 18x2y2÷24xy2=1824×x2x×y2y218x^2y^2 \div 24xy^2 = \frac{18}{24} \times \frac{x^2}{x} \times \frac{y^2}{y^2}

Simplify further: 34×x=3x4\frac{3}{4} \times x = \frac{3x}{4}

4) x2(2xz)(4z2)x^2(-2xz)(4z^2)

Multiply the terms:

  • First, multiply (2xz)(-2xz) and 4z24z^2: (2×4)×(xz×z2)=8xz3(-2 \times 4) \times (xz \times z^2) = -8xz^3
  • Then multiply by x2x^2: x2×(8xz3)=8x3z3x^2 \times (-8xz^3) = -8x^3z^3 The simplified expression is: 8x3z3-8x^3z^3

5) 5t+3r+9t10r85t + 3r + 9t - 10r - 8

Combine like terms:

  • 5t+9t=14t5t + 9t = 14t
  • 3r10r=7r3r - 10r = -7r

The simplified expression is: 14t7r814t - 7r - 8

6) (3ab)(3ab2)(3a2b3)(-3ab)(-3ab^2)(-3a^2b^3)

Multiply the terms: (3ab)×(3ab2)×(3a2b3)(-3ab) \times (-3ab^2) \times (-3a^2b^3) Multiply coefficients: (3)×(3)×(3)=27(-3) \times (-3) \times (-3) = -27 Multiply variables: a×a×a2=a4andb×b2×b3=b6a \times a \times a^2 = a^4 \quad \text{and} \quad b \times b^2 \times b^3 = b^6 The simplified expression is: 27a4b6-27a^4b^6

7) 5x7y4z313xy2z\frac{-5x^7y^4z^3}{13xy^2z}

Simplify by canceling out common terms: 5x7y4z313xy2z=513×x7x×y4y2×z3z\frac{-5x^7y^4z^3}{13xy^2z} = \frac{-5}{13} \times \frac{x^7}{x} \times \frac{y^4}{y^2} \times \frac{z^3}{z} Simplify further: 5x6y2z213\frac{-5x^6y^2z^2}{13}

8) (3pq2r3)(13q4r)(3pq^2r^3)(\frac{1}{3q^4r})

Simplify by multiplying: 3pq2r3×13q4r=33×p×q2q4×r3r3pq^2r^3 \times \frac{1}{3q^4r} = \frac{3}{3} \times p \times \frac{q^2}{q^4} \times \frac{r^3}{r} Simplify further: p×1q2×r2=pr2q2p \times \frac{1}{q^2} \times r^2 = \frac{pr^2}{q^2}


These are the simplified forms of each expression! Let me know if you'd like a detailed explanation of any step or have further questions.

Here are 5 follow-up questions to deepen your understanding:

  1. How do you determine like terms when combining algebraic expressions?
  2. What properties of exponents did we use in simplifying the expressions?
  3. How can factoring help in simplifying algebraic expressions?
  4. What strategies can be used to simplify fractions involving variables?
  5. What is the importance of canceling terms in rational expressions?

Tip: Always look for common factors or terms that can be grouped or canceled out to simplify complex expressions efficiently.

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Math Problem Analysis

Mathematical Concepts

Algebra
Simplification of expressions
Exponent rules
Factoring

Formulas

Combining like terms: ax + bx = (a+b)x
Exponent rules: a^m * a^n = a^(m+n), a^m / a^n = a^(m-n)

Theorems

Distributive Property: a(b + c) = ab + ac
Exponent Properties

Suitable Grade Level

Grades 8-10