Which property of exponents was used to generate the equivalent expression 3^14 from 3^5 / 3^−9?(1 point) Responses only the Property of Negative Exponents only the Property of Negative Exponents the Property of Negative Exponents and the Product Rule of Exponents the Property of Negative Exponents and the Product Rule of Exponents only the Power Rule of Exponents only the Power Rule of Exponents only the Product Rule of Exponents.
The property of negative exponents and the product rule of exponents were used to generate the equivalent expression 3^14 from 3^5 / 3^−9.
how
To see how the property of negative exponents and the product rule of exponents were used, let's break down the original expression:
3^5 / 3^−9
The property of negative exponents states that any number raised to a negative exponent is equal to its reciprocal raised to the positive exponent. So we can rewrite the expression as:
3^5 / (1 / 3^9)
Now, applying the product rule of exponents, which states that when you divide two numbers with the same base, you subtract the exponents, we have:
3^5 * 3^9 (the reciprocal is now in the numerator)
Simplifying further using the power rule of exponents, which states that when you multiply two numbers with the same base, you add the exponents, we get:
3^(5 + 9)
This simplifies to:
3^14
So, in total, we used the property of negative exponents (to convert the negative exponent to a positive exponent of the reciprocal) and the product rule of exponents (to combine the two expressions with the same base).
The property of negative exponents was used to generate the equivalent expression 3^14 from 3^5 / 3^−9.
To determine which property of exponents was used to generate the equivalent expression 3^14 from 3^5 / 3^−9, we can review the different exponent properties.
The three main properties of exponents are the Power Rule, Product Rule, and Negative Exponents Rule. Let's break them down:
1. Power Rule: This rule states that when you have an exponent raised to another exponent, you multiply the exponents. For example, (a^m)^n = a^(m*n).
2. Product Rule: This rule states that when you have two bases with the same exponent being multiplied, you can simply multiply the bases and keep the exponent the same. For example, a^m * b^m = (a * b)^m.
3. Negative Exponents Rule: This rule states that when you have an exponent with a negative sign, you can convert it to a positive exponent by moving the base to the opposite side of the fraction and changing the sign of the exponent. For example, a^(-m) = 1 / a^m.
Now let's analyze the given expression: 3^5 / 3^(-9).
To simplify this expression, we can use the Negative Exponents Rule. By moving 3^(-9) to the denominator and changing the sign of the exponent, it becomes 1 / 3^9. So now we have 3^5 multiplied by 1 / 3^9.
Next, we can use the Product Rule to combine the bases. Since we have 3^5 multiplied by 1 / 3^9, we can rewrite it as (3^5 * 1) / 3^9.
Finally, evaluating the numerator (3^5 * 1) gives us 3^5. And the denominator remains as 3^9.
Therefore, the property of exponents used to generate the equivalent expression 3^14 from 3^5 / 3^(-9) is the Product Rule of Exponents and the Property of Negative Exponents.