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Chapter 6: Problem 101

Simplify each expression using the Product Property for Exponents. $$q^{27} \cdot q^{15}$$

Short Answer

Expert verified
q^{42}

Step by step solution

01

- Identify the bases and exponents

In the expression, identify the base and the exponents. The given expression is \(q^{27} \times q^{15}\). Here, the base is \(q\), and the exponents are 27 and 15.
02

- Apply the Product Property of Exponents

The Product Property of Exponents states that when multiplying like bases, the exponents should be added. So, for \(a^m \times a^n = a^{m+n}\), apply this property: \(q^{27} \times q^{15} = q^{27 + 15}\).
03

- Add the exponents

Add the exponents: \(27 + 15\). This results in 42.
04

- Write the simplified expression

Combine the base with the new exponent: \(q^{42}\). The simplified form of the expression is \(q^{42}\).

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Key Concepts

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

exponents
In mathematics, an exponent represents the number of times a number, known as the base, is multiplied by itself. For instance, in the expression \(2^3\), the number 2 is the base, and 3 is the exponent. This expression means that 2 should be multiplied by itself three times: \(2 \times 2 \times 2 = 8\). Exponents are a shorthand way to denote repeated multiplication. They are incredibly useful in simplifying mathematical expressions.
base
The base in an exponential expression is the number that gets multiplied when raised to a power. For example, in \(q^{27} \times q^{15}\), the base is \(q\). Irrespective of the value of the exponents, the base remains the same in expressions following the Product Property of Exponents. Understanding what the base is helps in recognizing that only the exponents will change when like bases are multiplied.
simplifying expressions
Simplifying expressions involves reducing them to their most basic form. When dealing with exponents, it means applying the rules such as the Product Property of Exponents to make expressions more manageable. For instance, \(q^{27} \times q^{15}\) can be simplified by adding the exponents together because the bases are the same. Following the steps of combining bases first and then calculating the resultant exponent helps streamline the process.
addition of exponents
The addition of exponents comes into play when using the Product Property of Exponents. This property states that if you multiply like bases, you add the exponents. For example, in \(a^m \times a^n = a^{m+n}\), both the exponents \(m\) and \(n\) are added while the base \(a\) remains the same. In our specific case of \(q^{27} \times q^{15}\), adding the exponents (27+15) results in \(q^{42}\). This addition is crucial for simplifying the expression effectively and accurately.

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Most popular questions from this chapter

Mixed Practice $$(4 y)\left(12 y^{7}\right) \div 8 y^{2}$$

Divide each polynomial by the monomial. $$\left(63 m^{4}-42 m^{3}\right) \div\left(-7 m^{2}\right)$$

Mixed Practice (a) \(\frac{z^{6}}{z^{5}}\) (b) \(\frac{z^{5}}{z^{6}}\)

Divide each polynomial by the monomial. $$\frac{412 z^{8}-48 z^{5}}{4 z^{3}}$$

Divide each polynomial by the binomial. $$\left(x^{2}-3 x-10\right) \div(x+2)$$
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