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Chapter 12: Problem 119

Simplify. $$ x^{2} \cdot x^{3} $$

Short Answer

Expert verified
\(x^5\)

Step by step solution

01

Recall the Exponent Rule for Multiplication

When multiplying exponents with the same base, add the exponents together. The rule can be written as \[a^m \times a^n = a^{m+n} \].
02

Identify the Base and Exponents

In the given expression \(x^2 \times x^3\), the base is \(x\) and the exponents are 2 and 3, respectively.
03

Apply the Exponent Rule

Use the exponent rule to add the exponents together: \[x^{2+3}\].
04

Simplify the Expression

Add the exponents to get: \[x^5\].

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Key Concepts

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

simplifying expressions
Simplifying expressions means making them easier to work with. You break them down into a simpler form. In algebra, you often simplify by combining like terms or using mathematical properties like exponent rules.

For example, to simplify the expression \(x^2 \cdot x^3 \), you apply the rules of exponents. This process makes calculations easier since you deal with fewer and simpler terms.

Always look for opportunities to combine and reduce expressions in algebra. It saves time and reduces mistakes.
multiplication of exponents
Multiplying exponents might look tricky, but it's simple with the right rule. When the bases are the same, add the exponents. The rule is: \(a^m \cdot a^n = a^{m+n}\).

So when you see something like \[x^2 \cdot x^3\], remember that both terms have the same base (x). Using the rule, you add the exponents: 2 and 3. This gives you: \[x^{2+3} = x^5\].
  • Start by identifying the base and exponents.
  • Add the exponents together.
  • Write the simplified expression.

Multiplying exponents simplifies several small numbers into one bigger exponent. This keeps your work neat and manageable.
algebra
Algebra concerns letters and symbols used to represent numbers and quantities in formulas and equations. It's a method to solve problems step-by-step.

For the exercise \(x^2\cdot x^3\), you are using algebra to simplify the expression using exponent rules.
  • First, understand what each part of the expression represents.
  • Second, apply the right rules (like exponent rules) to simplify.
  • Lastly, arrive at the simplified solution
Algebra helps you solve problems logically. Using algebraic rules and properties, you can simplify even complicated expressions into manageable forms.

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Most popular questions from this chapter

Solve. $$ \log _{x} 11=\frac{1}{2} $$

Given \(\log _{b} 3=0.792 \text { and } \log _{b} 5=1.161\). If possible, use the properties of logarithms to calculate numerical values for each of the following. $$\log _{b} \frac{1}{5}$$

Express as an equivalent expression that is a sum or a difference of logarithms and, if possible, simplify. $$\log _{a} \sqrt{1-s^{2}}$$

Express as an equivalent expression that is a single logarithm and, if possible, simplify. $$\log _{a} \frac{a}{\sqrt{x}}-\log _{a} \sqrt{a x}$$

Simplify. $$ \frac{x^{12}}{x^{4}} $$
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