By Fatskills Exam Guides Team — the exam nerds behind 28,500+ quizzes and 2.1M practice questions across 500+ global exams.
The laws of exponents are a set of rules that govern how exponential expressions with the same base are combined. These rules allow us to simplify complex expressions and solve equations involving exponents.
Exponents are used extensively in various fields, including physics, engineering, economics, and computer science. For example, in physics, the exponential decay of radioactive materials is modeled using the half-life formula, which relies on the laws of exponents. In engineering, the design of electronic circuits involves the use of exponential functions to model the behavior of electrical components.
The product of powers rule states that when multiplying two exponential expressions with the same base, we add the exponents. Mathematically, this can be represented as: $$a^m \cdot a^n = a^{m+n}$$
The quotient of powers rule states that when dividing two exponential expressions with the same base, we subtract the exponents. Mathematically, this can be represented as: $$\frac{a^m}{a^n} = a^{m-n}$$
The power of a power rule states that when raising an exponential expression to a power, we multiply the exponents. Mathematically, this can be represented as: $$(a^m)^n = a^{m \cdot n}$$
The negative exponent rule states that any non-zero number raised to a negative power can be rewritten as the reciprocal of the number raised to the positive power. Mathematically, this can be represented as: $$a^{-n} = \frac{1}{a^n}$$
The zero exponent rule states that any non-zero number raised to the power of zero is equal to 1. Mathematically, this can be represented as: $$a^0 = 1$$
To approach problems involving the laws of exponents, follow these steps:
Simplify the expression: $2^3 \cdot 2^4$
Using the product of powers rule, we can rewrite the expression as: $$2^3 \cdot 2^4 = 2^{3+4} = 2^7$$
The simplified expression is $2^7$.
This result means that the product of $2^3$ and $2^4$ is equal to $2^7$.
Simplify the expression: $\frac{3^5}{3^2}$
Using the quotient of powers rule, we can rewrite the expression as: $$\frac{3^5}{3^2} = 3^{5-2} = 3^3$$
The simplified expression is $3^3$.
This result means that the quotient of $3^5$ and $3^2$ is equal to $3^3$.
Simplify the expression: $(2^3)^4$
Using the power of a power rule, we can rewrite the expression as: $$(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$$
The simplified expression is $2^{12}$.
This result means that the fourth power of $2^3$ is equal to $2^{12}$.
What is the simplified expression for $2^3 \cdot 2^4$? A) $2^6$ B) $2^7$ C) $2^5$ D) $2^9$
B) $2^7$
Using the product of powers rule, we can rewrite the expression as $2^{3+4} = 2^7$.
A) $2^6$ is the result of adding the exponents, but it's not the correct result. C) $2^5$ is the result of multiplying the exponents, but it's not the correct result. D) $2^9$ is the result of adding the exponents and then multiplying by 2, but it's not the correct result.
What is the simplified expression for $\frac{3^5}{3^2}$? A) $3^3$ B) $3^7$ C) $3^1$ D) $3^9$
A) $3^3$
Using the quotient of powers rule, we can rewrite the expression as $3^{5-2} = 3^3$.
B) $3^7$ is the result of adding the exponents, but it's not the correct result. C) $3^1$ is the result of subtracting the exponents, but it's not the correct result. D) $3^9$ is the result of adding the exponents and then multiplying by 3, but it's not the correct result.
What is the simplified expression for $(2^3)^4$? A) $2^{12}$ B) $2^5$ C) $2^9$ D) $2^1$
A) $2^{12}$
Using the power of a power rule, we can rewrite the expression as $2^{3 \cdot 4} = 2^{12}$.
B) $2^5$ is the result of multiplying the exponents, but it's not the correct result. C) $2^9$ is the result of adding the exponents, but it's not the correct result. D) $2^1$ is the result of subtracting the exponents, but it's not the correct result.
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