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Chapter 6: Problem 45

Write with a single exponent. $$ \frac{(a+b)^{5}}{(a+b)^{2}} $$

Short Answer

Expert verified
Question: Simplify the expression: $$\frac{(a+b)^{5}}{(a+b)^{2}}$$ Answer: The simplified expression is: $$(a+b)^{3}$$

Step by step solution

01

Identify the base and exponents for the given expression

The given expression is $$ \frac{(a+b)^{5}}{(a+b)^{2}} $$ Here, the base is (a+b), the exponent in the numerator is 5, and the exponent in the denominator is 2.
02

Apply the exponent rule for division

To write the given expression with a single exponent, we will use the division rule for exponents, which states that when we divide two expressions with the same base, we subtract their exponents. So, we have: $$ (a+b)^m \div (a+b)^n = (a+b)^{m-n} $$
03

Substitute the values in the rule

Now, we will substitute the values of the exponents for m (numerator) and n (denominator) into the rule: $$ \frac{(a+b)^{5}}{(a+b)^{2}} = (a+b)^{5-2} $$
04

Calculate the new exponent

Now, we will calculate the new exponent: $$ 5-2 = 3 $$
05

Write the final expression

Finally, substitute the new exponent back into the expression: $$ (a+b)^{3} $$ So, the given expression with a single exponent is: $$ \frac{(a+b)^{5}}{(a+b)^{2}} = (a+b)^{3} $$

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Key Concepts

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

Division of Exponents
When you're dealing with exponents, there are specific rules that can make your life easier. One of these important rules is the division of exponents. Let's say you have two expressions that are being divided and they have the same base. Like in our example, both the numerator and the denominator have the base \((a+b)\). The rule tells us that we can simply subtract the exponent in the denominator from the exponent in the numerator. This rule is expressed as:
  • If you have \((a+b)^m \div (a+b)^n\), it simplifies to \((a+b)^{m-n}\).
This means, when dividing with the same base, you do not multiply the exponents; you subtract them. It鈥檚 a handy rule and cuts down a lot of work in simplifying expressions.
Simplifying Expressions
Simplifying expressions is all about making them as straightforward as possible. Applying the division rule we just learned, we can make expressions less complex. For example, applying this rule to our exercise by subtracting the exponents:
  • We have \(5 - 2 = 3\), so \((a+b)^3\) is our simplified result.
Making expressions simpler helps in reducing chances of error in further calculations. Plus, it's easier to understand what you鈥檙e working with when the expression is clean and clear. Always look for opportunities to simplify wherever possible.
Algebraic Manipulation
Algebraic manipulation involves a series of techniques to rearrange or simplify expressions. This is crucial in solving equations and altering expressions for clarity or simplicity. With our exercise, understanding and using the exponent rules demonstrate a form of algebraic manipulation.
  • You identify common bases and use the properties of exponents to rewrite the expression.
  • This kind of manipulation is helpful for solving algebraic equations, optimizing expressions, and recognizing patterns.
The ability to manipulate algebraic expressions efficiently is a powerful skill, especially in solving complex problems. Mastery over these basic rules helps in developing a deeper understanding of algebra.

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Most popular questions from this chapter

Write each expression without parentheses. Assume all variables are positive. $$ \left(\frac{3}{w^{4}}\right)^{4} $$

Write with a single exponent. $$ B^{a} B^{a+1} $$

Without a calculator, decide whether the quantities are positive or negative. $$ -4^{3} $$

Write each expression without parentheses. Assume all variables are positive. $$ \left(\frac{c^{3}}{d}\right)^{4} $$

Write each expression without parentheses. Assume all variables are positive. $$ (2 p)^{5} $$
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