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Chapter 7: Problem 3

Simplify each expression. In each exercise, all variables are positive. \(x^{3} \cdot x^{4}\)

Short Answer

Expert verified
The simplified expression is \(x^7\).

Step by step solution

01

Understand the Properties of Exponents

The problem asks us to simplify the expression \(x^3 \cdot x^4\). The first step is recognizing that when you multiply powers with the same base, you can add their exponents. This follows the property \(a^m \cdot a^n = a^{m+n}\).
02

Apply the Multiplication of Exponents Rule

Using the rule from Step 1, we add the exponents of the same base \(x\), giving us: \(x^3 \cdot x^4 = x^{3+4}\).
03

Simplify the Expression

Perform the addition of the exponents: \(3 + 4 = 7\). This simplifies our expression to \(x^7\).

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Key Concepts

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

Multiplying Powers with Same Base
When dealing with powers that share the same base, a useful property greatly simplifies the process of multiplication. If you see an expression like \(x^3 \cdot x^4\), both terms share the base "\(x\)". The rule for multiplying powers with the same base states that you should add their exponents.
This property can be expressed as:
  • \(a^m \cdot a^n = a^{m+n}\)
With this knowledge, combining two powers like \(x^3\) and \(x^4\) becomes straightforward. You just add the exponents, resulting in \(x^{3+4}\), simplifying further to \(x^7\). This purple rule stays consistent across numbers, variables, or any other powers as long as they share the same base.
Simplifying Expressions
Simplifying expressions is a core skill in algebra that involves reducing expressions to their most basic form without changing their value. When you're given an expression involving multiplication of powers, like \(x^3 \cdot x^4\), simplification involves making it as clean and manageable as possible.
  • First, identify any properties of exponents that can be used, like adding exponents when multiplying powers with the same base.
  • Next, carry out any arithmetic operations, like addition of exponents, to simplify the expression.
  • Finally, ensure that the expression is in its simplest form, with no further operations needed.
By transforming \(x^3 \cdot x^4\) into \(x^7\), it becomes easier to work with this expression in further computations or evaluations.
Addition of Exponents
The addition of exponents is a direct result of the multiplying properties of powers. When you multiply powers with the same base, you add the exponents. Let's dissect this a bit more: An exponent tells you how many times to multiply a number by itself.
For instance:
  • \(x^3\) means \(x\) multiplied by itself three times: \(x \cdot x \cdot x\).
  • \(x^4\) means \(x\) multiplied by itself four times: \(x \cdot x \cdot x \cdot x\).
So, when you multiply \(x^3\) and \(x^4\), you're effectively multiplying \(x\) by itself a total of seven times, illustrating why the rule \(a^m \cdot a^n = a^{m+n}\) holds.
Understanding the rationale behind adding exponents enables you to confidently simplify expressions involving powers, such as finding \(x^7\) from \(x^3 \cdot x^4\).

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Most popular questions from this chapter

In \(11-16,\) use the formula \(A=A_{0}\left(1+\frac{r}{n}\right)^{n t}\) to find the missing variable to the nearest hundredth. $$ A=400, r=5 \%, n=4, t=3 $$

In \(35-63,\) write each expression with only positive exponents and express the answer in simplest form. The variables are not equal to zero. $$ \left(\frac{3 m^{-3}}{2 n^{-2}}\right)^{-3} $$

In \(35-63,\) write each expression with only positive exponents and express the answer in simplest form. The variables are not equal to zero. $$ \left(a^{5} b^{-5} c^{-4}\right)^{-3} $$

In \(64-75,\) write each quotient as a product without a denominator. The variables are not equal to zero. $$ \frac{3}{x^{4}} $$

In \(58-73\) , write each power as a radical expression in simplest form. The variables are positive numbers. $$ \left(16 a^{5} b^{6}\right)^{\frac{1}{4}} $$
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