Apply the Properties of Integer Exponents to generate equivalent expressions to (4^3 ⋅ 7^−2)^−5 7^9 ⋅ 4^−13 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
(4^3 ⋅ 7^−2)^−5 7^9 ⋅ 4^−13=?
To generate equivalent expressions with only positive exponents, we can use the following properties of integer exponents:
1. (a^m)^n = a^(m*n)
2. a^(-m) = 1/(a^m)
3. a^m * a^n = a^(m+n)
4. (a/b)^m = a^m/b^m
Using these properties, let's simplify the given expression:
(4^3 ⋅ 7^(-2))^(-5) 7^9 ⋅ 4^(-13)
= (64 ⋅ 1/49)^(-5) 7^9 ⋅ 1/(4^13) (Using properties 2 and 4 to make the exponents positive)
= (64/49)^(-5) 7^9/4^13 (Simplifying the expression inside the parentheses)
= (49/64)^5 7^9/4^13 (Taking the reciprocal of 64/49 to make the exponent positive)
= 49^5/(64^5) 7^9/4^13 (Using property 3 to combine the two bases)
= (7^2)^5/(4^2)^5 7^9/4^13 (Writing 49 and 64 as powers of 7 and 4 respectively)
= 7^10/4^10 7^9/4^13 (Using property 1 to multiply the exponents)
= 7^(10+9)/4^(10+13) (Using property 3 to combine the exponents)
= 7^19/4^23 (Simplifying the exponents)
Therefore, the equivalent expression is 7^19/4^23.
To generate equivalent expressions with only positive exponents, we can use the following properties of integer exponents:
1. Power of a Product: (ab)^n = a^n * b^n
2. Power of a Quotient: (a/b)^n = a^n / b^n
3. Negative Exponent: a^(-n) = 1 / a^n
Let's simplify the expression step-by-step:
Step 1: Apply the Power of a Product property inside the parentheses:
(4^3 ⋅ 7^−2)^−5 = 4^(3*-5) * 7^(-2*-5)
Step 2: Simplify the exponents:
= 4^(-15) * 7^10
Step 3: Apply the Negative Exponent property to 4^(-15):
= 1 / 4^15 * 7^10
Step 4: Simplify the expression outside the parentheses:
1 / 4^15 * 7^10 * 7^9 * 4^(-13)
Step 5: Combine the like bases (4 and 7) using the Power of a Product property:
= 1 / (4^15 * 4^(-13)) * (7^10 * 7^9)
Step 6: Simplify the exponents for each base:
= 1 / (4^(15+(-13))) * 7^(10+9)
Step 7: Calculate the exponents:
= 1 / 4^2 * 7^19
Step 8: Simplify the expression further:
= 1 / 16 * 7^19
Therefore, the simplified expression is 1/16 * 7^19.
To generate equivalent expressions with only positive exponents, we can use the properties of integer exponents. Let's break it down step by step:
Step 1: Apply the power of a product property.
(a^m * b^n)^p = a^(m*p) * b^(n*p)
Applying this property to our expression:
(4^3 * 7^(-2))^(-5) * 7^9 * 4^(-13)
= 4^(3*(-5)) * 7^((-2)*(-5)) * 7^9 * 4^(-13)
= 4^(-15) * 7^10 * 7^9 * 4^(-13)
Step 2: Simplify the exponents.
= 4^(-15) * 7^(10+9) * 4^(-13)
= 4^(-15) * 7^19 * 4^(-13)
Step 3: Apply the power of a power property.
(a^m)^n = a^(m*n)
Applying this property to our expression:
4^(-15) * 7^19 * 4^(-13) = 4^((-15)*(-1)) * 7^19 * 4^(-13)
= 4^15 * 7^19 * 4^(-13)
Step 4: Apply the power of a product property again.
a^m * a^n = a^(m+n)
Applying this property to our expression:
4^15 * 7^19 * 4^(-13) = 4^15 * 4^(-13) * 7^19
= 4^(15+(-13)) * 7^19
= 4^2 * 7^19
= 16 * 7^19
So, the equivalent expression is 16 * 7^19. This cannot be simplified further, as there are no remaining exponents.