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Chapter 10: Problem 2

Simplify. $$x^{3} x^{12} x$$

Short Answer

Expert verified
The simplified form is \(x^{16}\).

Step by step solution

01

Identify the exponents

The expression involves the multiplication of three terms: \(x^3\), \(x^{12}\), and \(x\). Recall that \(x\) is the same as \(x^1\).
02

Combine the exponents

When multiplying terms with the same base, add the exponents. The expression simplifies as follows: \[x^3 \times x^{12} \times x^1 = x^{3+12+1}\]
03

Add the exponents

Calculate the sum of the exponents: \[3 + 12 + 1 = 16\]
04

Write the simplified expression

Combine the results from the previous step to get the simplified form of the expression: \[x^{16}\]

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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.
The goal is to express the given mathematical expression in its simplest form.
This often involves combining like terms, reducing fractions, or using exponent rules.
For example, in the expression we are simplifying, we use the rule that when multiplying like bases, we add the exponents.
Here鈥檚 the expression we started with: \(x^3 \times x^{12} \times x\).
Notice all terms have the base 'x', so our task is to combine the exponents.
By knowing basic rules and strategies, such as those for exponents, we can simplify algebraic expressions efficiently.
multiplication of exponents
The multiplication of exponents is a key concept in algebra.
When you multiply terms with the same base, you simply add the exponents.
This rule arises from how exponents represent repeated multiplication.
For example, in our problem, we had \(x^3 \times x^{12} \times x^1\).
Applying the multiplication rule, we add the exponents: \(3+12+1=16\).
Therefore, the expression becomes \(x^{16}\).
The general formula for multiplying exponents with the same base is: \[a^m \times a^n = a^{m+n}\].
This lets you simplify expressions and solve problems more easily and accurately.
basic algebra
Basic algebra involves understanding fundamental operations and rules.
This includes working with variables, constants, and exponents.
Algebra allows us to solve equations and simplify expressions like the one in this exercise.
In our example, the variable 'x' appears raised to different powers.
We applied the exponent multiplication rule to combine these terms effectively.
Algebra also teaches us to look for patterns and use properties of operations to simplify and solve problems.
Mastering these basic principles aids in tackling more complex algebraic challenges quickly and confidently.

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Most popular questions from this chapter

Simplify. Write the answers with positive exponents only. $$w^{4} w^{-7}$$

Simplify. Write the answers with positive exponents only. $$\left(k^{-7}\right)^{-2}$$

Subtract the polynomials. $$\left(9 p^{3}-4 p^{2}+2 p\right)-\left(2 p^{3}+6 p^{2}-3 p\right)$$

Write the opposite of the polynomial. $$9 x^{2}-2 x-3$$

Perform the indicated operations. $$\left(-5.1 y^{2}+4.8 y+2.3\right)+\left(-1.1 y^{2}-8.9 y+3.0\right)$$
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