Problem 632 Simplify each expression. $$\l... [FREE SOLUTION]

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Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 632

Simplify each expression. $$\left(8 x y^{3}\right)\left(-6 x^{4} y^{6}\right)$$

Short Answer

Expert verified

-48 x^5 y^9

Step by step solution

Multiply the coefficients

First, multiply the numerical coefficients 8 and -6: \[ 8 \times (-6) = -48 \]

Multiply the x terms

Next, multiply the terms involving x. Use the rule for multiplying exponents with the same base \[ x^a \times x^b = x^{a+b} \]. So, \[ x \times x^4 = x^{1+4} = x^5 \]

Multiply the y terms

Then, multiply the terms involving y. Again, use the rule for multiplying exponents with the same base \[ y^a \times y^b = y^{a+b} \]. So, \[ y^3 \times y^6 = y^{3+6} = y^9 \]

Combine all the results

Now, combine all the results from the previous steps: \[ -48 x^5 y^9 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves combining like terms and applying mathematical operations. It makes expressions easier to understand and work with. For example, consider $\big(8xy^3\big)\big(-6x^4y^6\big)$. Begin by writing the expression clearly and then focus on each separate part. Combining like terms helps simplify the equation to its most basic form. Use the rules of exponents to handle the variables and their respective powers. Keep the process organized and methodical to avoid mistakes.

Rules of Exponents

The rules of exponents are guidelines for performing operations involving powers. These rules simplify working with expressions where variables have exponents.
- Multiplying exponents: When multiplying terms with the same base, add their exponents. The general rule is $a^m \times a^n = a^{m+n}$.
Example: \ { x \times x^4 = x^{1+4} = x^5 } \
- Zero exponent rule: Any number raised to the power of zero is 1, so $a^0 = 1$.
These basic rules are integral to simplifying and solving algebraic expressions involving exponents.

Multiplication of Coefficients

Multiplying coefficients means multiplying the numerical parts of the terms in an expression separately from the variables. This is usually the first step in simplifying expressions.
In the given expression, $\big(8xy^3\big)\big(-6x^4y^6\big)$, the coefficients are 8 and -6. Multiply them together: \ { 8 \times (-6) = -48 } \
This step simplifies the numerical part of the expression. Once the coefficients are multiplied, proceed to handle the variables using the rules of exponents. This will help ensure the expression is fully simplified and manageable.

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91影视

Simplify each expression. $$\left(8 x y^{3}\right)\left(-6 x^{4} y^{6}\right)$$

Short Answer

Step by step solution

Multiply the coefficients

Multiply the x terms

Multiply the y terms

Combine all the results

Key Concepts

Simplifying Algebraic Expressions

Rules of Exponents

Multiplication of Coefficients

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