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Chapter 5: Problem 107

Multiply. $$ \left(-m^{2}\right)\left(m^{5}\right) $$

Short Answer

Expert verified
-m^7

Step by step solution

01

Identify the bases

Recognize that both terms have the same base, which is the variable \( m \).
02

Identify the exponents

The first term \( -m^2 \) has an exponent of 2, and the second term \( m^5 \) has an exponent of 5.
03

Use the Power of a Product Property

When multiplying terms with the same base, add their exponents: \[ -m^2 \times m^5 = -m^{2+5} \]
04

Simplify the expression

Add the exponents: \[ -m^{2+5} = -m^7 \]

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Key Concepts

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

powers of a product property
To understand the 'Powers of a Product' property, it's essential to know how this principle works for multiplying powers with the same base.
When you multiply two expressions that have the same base, you can add their exponents together. This is a very handy property. For instance, when you see something like \(-m^2 \times m^5\), you know you can simplify this by adding the exponents.
This step makes calculating powers much easier!
You just need to remember that you are working with the same base; in this case, the base is \( m \).
same base multiplication
In algebra, 'Same Base Multiplication' is a straightforward yet important rule.
To apply this rule, identify that the expressions being multiplied share the same base. In our example, both expressions \(-m^2\) and \(m^5\) have the base \( m \).
Once you identify the base, you can confidently add the exponents. Remember not to multiply the exponents, only add them.
For our problem, the base \( m \) stays the same, and we simply add the exponents 2 and 5 together to form \( m^7 \).
algebraic expressions
Algebraic expressions involve constants, variables, and their combinations using various operations.
Here we have an algebraic expression involving multiplication of powers: \(-m^2\) and \(m^5\).
Variables like \( m \) can have various exponents and can be multiplied according to algebraic properties. Simply follow the rules like 'Powers of a Product' and 'Same Base Multiplication' to simplify such expressions.
Once you understand these rules, handling algebraic expressions becomes much easier and more intuitive.

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Most popular questions from this chapter

The special product $$ (x+y)(x-y)=x^{2}-y^{2} $$ $$ \text { can be used to perform some multiplication problems. Here are two examples.} $$ $$ \begin{aligned} 51 \times 49 &=(50+1)(50-1) \\ &=50^{2}-1^{2} \\ &=2500-1^{2} \\ &=2499 \end{aligned} \quad | \begin{aligned} 102 \times 98 &=(100+2)(100-2) \\ &=100^{2}-2^{2} \\ &=10,000-4 \\ &=9996 \end{aligned} $$ Once these patterns are recognized, multiplications of this type can be done mentally. Use this method to calculate each product mentally. $$ 101 \times 99 $$

Evaluate. $$ 49 \div 10,000 $$

Write each product as a sum of terms. Write answers with positive exponents only. Simplify each term. See Section 1.8 $$ \frac{1}{4 y}\left(y^{4}+6 y^{2}+8\right) $$

If it costs \(\$ 15\) to rent a chain saw, plus \(\$ 2\) per day, the binomial \(2 x+15\) gives the cost to rent the chain saw for \(x\) days. Evaluate this polynomial for \(x=6 .\) Use the result to fill in the blanks: If the saw is rented for ___\(-\) days, the cost is ___.

Perform each division using the "long division" process. $$ \frac{8 k^{4}-12 k^{3}-2 k^{2}+7 k-6}{2 k-3} $$
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