The product to power property is a fundamental rule in exponent arithmetic that states the product of two numbers raised to the same power is equal to the original numbers raised to that power, multiplied together. This property simplifies the calculation of expressions involving exponents and powers.
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The product to power property states that $a^m \cdot a^n = a^{m+n}$, where $a$ is the base and $m$ and $n$ are the exponents.
This property allows you to simplify expressions by combining the exponents when multiplying numbers raised to the same base.
The product to power property is useful for simplifying expressions involving multiplication of powers with the same base, as it reduces the number of steps required.
Understanding the product to power property is crucial for efficiently solving problems related to exponent arithmetic, such as those found in the topic of 6.2 Use Multiplication Properties of Exponents.
Applying the product to power property correctly is essential for manipulating and simplifying algebraic expressions involving exponents.
Review Questions
Explain the product to power property and how it can be used to simplify expressions involving exponents.
The product to power property states that when you multiply numbers raised to the same base, you can add the exponents. Mathematically, this is represented as $a^m \cdot a^n = a^{m+n}$, where $a$ is the base and $m$ and $n$ are the exponents. This property allows you to simplify expressions by combining the exponents, which reduces the number of steps required. For example, if you have $3^2 \cdot 3^4$, you can use the product to power property to rewrite this as $3^{2+4} = 3^6$, simplifying the expression.
Describe how the product to power property relates to the topic of 6.2 Use Multiplication Properties of Exponents.
The product to power property is a crucial concept within the topic of 6.2 Use Multiplication Properties of Exponents. This property allows you to simplify expressions involving the multiplication of powers with the same base, which is a common task in this chapter. Understanding and correctly applying the product to power property is essential for efficiently solving problems related to exponent arithmetic, as it enables you to combine exponents and reduce the complexity of the expressions. Mastering this property will help you successfully navigate the various multiplication rules and properties covered in the 6.2 topic.
Evaluate the expression $2^3 \cdot 2^5$ using the product to power property, and explain the steps involved.
To evaluate the expression $2^3 \cdot 2^5$ using the product to power property, we can apply the rule that $a^m \cdot a^n = a^{m+n}$, where $a$ is the base and $m$ and $n$ are the exponents. In this case, the base is $2$, and the exponents are $3$ and $5$. Substituting these values, we get $2^3 \cdot 2^5 = 2^{3+5} = 2^8 = 256$. By using the product to power property, we were able to combine the exponents and simplify the expression, arriving at the final result of $256$.