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Chapter 7: Problem 12

Simplify each of the following as completely as possible. $$t^{5} t^{2} t^{8}$$

Short Answer

Expert verified
The simplified form is \(t^{15}\).

Step by step solution

01

- Recall the Product of Powers Property

The property states that when multiplying powers with the same base, add the exponents: \(a^{m} \times a^{n} = a^{m+n}\).
02

- Apply the Product of Powers Property

Identify the exponents of the base \(t\) and add them together: \(t^{5} \times t^{2} \times t^{8} = t^{5+2+8}\).
03

- Simplify the Exponent

Add the exponents: \(5 + 2 + 8 = 15\).
04

- Write the Simplified Expression

Combine the base with the simplified exponent: \(t^{15}\).

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Key Concepts

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

Product of Powers Property
When combining multiple terms of the same base with exponents, one powerful tool to simplify the expression is the Product of Powers Property. This property states: \(a^{m} \times a^{n} = a^{m+n}\).
Essentially, it means that you add the exponents when you multiply terms with the same base.

Let's apply this to our example: \(t^{5} t^{2} t^{8}\). Here, the base is \(t\), and the exponents are 5, 2, and 8 respectively.
So, following the property, you simply add the exponents together: \(5 + 2 + 8\). This results in the combined expression: \(t^{15}\).
Understanding this property makes dealing with exponents more straightforward and easier to manage.
Understanding Exponents
Exponents are a way to express repeated multiplication of a number by itself. For example, \(t^{3}\) means \(t \times t \times t\).
Exponents can have multiple bases and values, and the rules for manipulating them like addition or multiplication remain consistent.

There are several rules for dealing with exponents, such as the Product of Powers Property we've discussed. Here鈥檚 a brief overview:
  • Multiplication: Add exponents \(a^{m} \times a^{n} = a^{m+n}\)
  • Division: Subtract exponents \(a^{m} 梅 a^{n} = a^{m-n}\)
  • Power of a Power: Multiply exponents \( (a^{m})^{n} = a^{m \times n}\)

Mastering these basic rules will help you handle any algebra problem involving exponents with ease.
Simplifying Expressions
Simplifying expressions involves reducing them to their simplest form without changing their value. This usually means combining like terms and using algebraic properties effectively.

In the context of exponents, simplifying means:
  • Combining terms using the Product of Powers Property
  • Reducing complex fractions where exponents are involved
  • Applying rules such as negative exponents or zero exponents

For our example, we start with \(t^{5} t^{2} t^{8} \), use the Product of Powers Property to add the exponents (\(5 + 2 + 8 = 15\)), and end with a simplified expression: \(t^{15}\).
Understanding and practicing these concepts will help you simplify expressions efficiently and accurately, leading to a clearer and more concise answer.

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Most popular questions from this chapter

Do your computations using scientific notation. The diameter of an atom might typically be 3.47 A. Write this number in scientific notation.

Do your computations using scientific notation. If one atom of iron has a mass of \(9.3 \times 10^{-23}\) gram, what is the mass of \(80,000\) atoms?

Solve for x. $$5-2 x \geq 17$$

Multiply out each of the following. As you work out the problems, identify those exercises that are either a perfect square or the difference of two squares. $$(2 a-5)^{2}$$

Convert each number into scientific notation. $$111.12$$
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