Which of the following is an equivalent expression to 5⁹ • 5^-¹³ with only positive exponents that has been generated by applying the properties of integer exponents?
A. 1/625
B. 5⁴
C. 5⁹/5^-¹³
D. 1/5⁴
To simplify the expression 5⁹ • 5⁻¹³, we can use the property of exponents that states: aⁿ • aᵐ = aⁿ⁺ᵐ.
Applying this property, we add the exponents of 5 together: 9 + (-13) = -4.
So, 5⁹ • 5⁻¹³ = 5⁻⁴.
To convert this expression into an equivalent expression with positive exponents, we can use the property that states: a⁻ⁿ = 1/aⁿ.
Applying this property, we get: 5⁻⁴ = 1/5⁴.
Therefore, the equivalent expression with only positive exponents is option D: 1/5⁴.
Which property of exponents was used to generate the equivalent expression 3¹⁴ from 3⁵/3^-⁹?
A. Only the power rule of exponents
B. The property of negative exponents and the product rule of exponents
C. Only the product rule of exponents
D. Only the property of negative exponents
To simplify the expression 3⁵ / 3⁻⁹, we can use the property of exponents that states: aⁿ / aᵐ = aⁿ⁻ᵐ.
Applying this property, we subtract the exponent of 3 in the denominator from the exponent in the numerator: 5 - (-9) = 5 + 9 = 14.
So, 3⁵ / 3⁻⁹ = 3¹⁴.
The property of negative exponents was used in this simplification.
Therefore, the correct answer is D. Only the property of negative exponents.
Which of the following is an equivalent expression to 15⁰•7^-²/(-4)^-³ with only positive exponents that has been generated by applying the properties of integer exponents?
A. 1/7²•(-4)^-³
B. (-4)³/7²
C. 4³/7²
D. 7²/(-4)³
To simplify the expression 15⁰ • 7⁻² / (-4)⁻³, we can use the property of exponents that states: a⁰ = 1 for any non-zero value of a.
So, 15⁰ simplifies to 1: 1 • 7⁻² / (-4)⁻³.
Next, we can use the property of negative exponents that states: a⁻ⁿ = 1 / aⁿ.
Applying this property to the expression, we get: 1 • 1 / (1 / (-4)³) = 1/1 • (-4)³ = (-4)³.
Therefore, the equivalent expression with only positive exponents is option B: (-4)³/7².
Which of the following is the equivalent expression to (15^-³•4⁷)⁰/4^-³ that has been generated by applying the properties of integer exponents?
A. 1•4^-³
B. 1/64
C. 60/4^-³
D. 64
To simplify the expression (15⁻³•4⁷)⁰ / 4⁻³, we can apply the property of exponents that states a⁰ = 1 for any non-zero value of a.
The expression (15⁻³•4⁷)⁰ simplifies to 1: 1 / 4⁻³.
Next, we can use the property of negative exponents that states a⁻ⁿ = 1 / aⁿ.
Applying this property to the expression, we get 1 / (1 / 4³) = 1 / 1 / 64 = 64.
Therefore, the equivalent expression is option D: 64.
Which of the following is an equivalent expression to (2^-³ • 9³)^-⁴/2⁹ • 9^-¹⁰ with only positive exponents, generated by applying the properties of integer exponents?
A. 2³/9²
B. 2^-7 • 9^-¹/2⁹ • 9^-¹⁰
C. 2³ • 9²
D. 2¹² • 9^-¹²/2⁹ • 9^-¹⁰
To simplify the expression (2⁻³ • 9³)⁻⁴/2⁹ • 9⁻¹⁰, we will apply the properties of integer exponents.
First, let's work with the expression inside the parentheses, (2⁻³ • 9³). Applying the property of negative exponents that states a⁻ⁿ = 1/aⁿ, we can rewrite this as 1/(2³ • 9³) = 1/(8 • 729) = 1/5832.
Now, let's substitute this value back into the original expression:
(1/5832)⁻⁴ / 2⁹ • 9⁻¹⁰
According to the property of negative exponents, (a/b)ⁿ = bⁿ/aⁿ, we can rewrite (1/5832)⁻⁴ as 5832⁴. Therefore, the expression becomes:
5832⁴ / 2⁹ • 9⁻¹⁰
Now, using the property of exponents that states aⁿ • aᵐ = aⁿ⁺ᵐ, we can combine the exponents of 5832⁴ / 2⁹:
5832⁴ / (2⁹ • 9⁻¹⁰)
At this point, we can simplify the expression as much as possible.
The correct equivalent expression with only positive exponents is option A: 2³/9².