Apply the properties of integer exponents to generate equivalent expressions to 3⁷ • 3^-⁹ with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
3⁷ • 3^-⁹=
To generate equivalent expressions with only positive exponents, we can use the properties of integer exponents.
First, let's apply the property that states a^m • a^n = a^(m+n):
3^7 • 3^(-9) = 3^(7 + (-9))
Next, simplify the exponent:
3^(-2)
We can apply another property that states a^(-n) = 1/a^n:
1/3^2
Simplifying further, we have:
1/9
Therefore, the expression 3⁷ • 3^-⁹ is equal to 1/9.
Apply the properties of integer exponents to generate equivalent expressions to 2²/2^-⁴ with only positive exponents, then solve the expression. Provide the solution as a whole number.
To generate equivalent expressions with only positive exponents, we can use the properties of integer exponents.
Let's first apply the property that states a^m / a^n = a^(m-n):
2^2 / 2^(-4) = 2^(2 - (-4)) = 2^(2 + 4) = 2^6
Therefore, the expression 2²/2^-⁴ is equivalent to 2^6.
To solve this expression as a whole number, we calculate 2^6:
2^6 = 2 × 2 × 2 × 2 × 2 × 2 = 64
Therefore, the solution to the expression 2²/2^-⁴ is 64.
Apply the properties of integer exponents to generate equivalent expressions to 4⁰ • 5^-²/(-2)^-³ with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
To generate equivalent expressions with only positive exponents, we can use the properties of integer exponents.
First, let's simplify each term separately.
4^0 = 1 (Any number raised to the power of 0 is equal to 1.)
5^(-2) = 1/5^2 = 1/25
(-2)^(-3) = 1/(-2)^3 = 1/(-2 × -2 × -2) = 1/(-8) = -1/8
Now let's substitute the simplified terms back into the expression:
4^0 • 5^-2 / (-2)^-3 = 1 • 1/25 / -1/8
Next, let's simplify the expression:
1 • 1/25 / -1/8 = 1/25 ÷ -1/8
To divide fractions, we can multiply by the reciprocal of the second fraction:
1/25 ÷ -1/8 = 1/25 × -8/1
Multiplying the numerators and denominators:
1/25 × -8/1 = -8/25
Therefore, the solution to the expression 4⁰ • 5^-² / (-2)^-³ is -8/25.
To generate equivalent expressions with only positive exponents, we can use the property of exponents which states that when multiplying powers with the same base, we add their exponents. Additionally, if we have a negative exponent, we can flip the fraction and change the sign of the exponent to make it positive.
Starting with 3^7 • 3^(-9), we can change the second term to have a positive exponent by flipping the fraction and changing the sign of the exponent:
3^7 • 1/(3^9)
Now, using the property of exponents, when we multiply the two terms, we add their exponents:
3^(7 + 9) • 1
Simplifying the exponent, we get:
3^16 • 1
Multiplying any number by 1 doesn't change it, so we can just ignore the 1:
3^16
Therefore, the expression 3^7 • 3^(-9) is equivalent to 3^16.
To solve the expression, we just need to evaluate 3^16:
3^16 = 43,046,721
So, the simplified fraction with no remaining exponents is 43,046,721/1.
To generate an equivalent expression with only positive exponents, we can use the properties of integer exponents. One of these properties states that for any integer a and any integers m and n, where n is not equal to zero, a^m • a^n = a^(m + n).
Applying this property to our expression, we have:
3^7 • 3^(-9)
Using the property mentioned above, we can add the exponents:
3^(7 + (-9))
Simplifying the exponent:
3^(-2)
Now we have an equivalent expression with a negative exponent. To convert it into an expression with a positive exponent, we can use the property that a^(-n) = 1 / a^n.
Applying this property to our expression, we have:
1 / 3^2
Calculating 3^2:
1 / 9
Therefore, the simplified fraction equivalent to 3^7 • 3^(-9) is 1/9.