Apply the properties of integer exponents to generate equivalent expressions to
(4³•7^-²)^-⁵/7⁹•4^-¹³ with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
To generate equivalent expressions with positive exponents, we can use the following properties of integer exponents:
1) Product of Powers Property: (a^m)*(a^n) = a^(m+n)
2) Quotient of Powers Property: (a^m)/(a^n) = a^(m-n)
3) Power of Power Property: (a^m)^n = a^(m*n)
4) Negative Exponent Property: a^(-n) = 1/(a^n)
Let's apply these properties to simplify the expression step by step:
(4³•7^-²)^-⁵/7⁹•4^-¹³
Using the Power of Power Property:
= (4^3)^-5 • (7^-2)^-5 / (7^9) • (4^-13)
Using the Negative Exponent Property:
= 4^(-3*5) • 7^(2*5) / 7^9 • 4^(-13)
Simplifying the exponents:
= 4^(-15) • 7^10 / 7^9 • 4^(-13)
Using the Quotient of Powers Property:
= 4^(-15 - (-13)) • 7^(10-9)
Simplifying further:
= 4^(-15 + 13) • 7^1
= 4^(-2) • 7
Using the Negative Exponent Property:
= 1/(4^2) • 7
= 1/16 • 7
Now, let's multiply and simplify the fraction:
= 7/16
So, the simplified fraction without any remaining exponents is 7/16.
To generate equivalent expressions with only positive exponents, we can use the properties of integer exponents as follows:
Step 1: Distribute the negative exponent to the numerator and denominator separately:
(4³ • 7^-²)^-⁵ / (7⁹ • 4^-¹³)
= (4³)^-⁵ • (7^-²)^-⁵ / (7⁹)^-⁵ • (4^-¹³)^-⁵
Step 2: Simplify the exponents by applying the negative exponent rule:
= 4^-¹⁵ • 7^¹⁰ / 7^-⁴⁵ • 4^²⁶⁵
Step 3: Simplify further by combining the exponents of the same base:
= 4^(-1⁵ + ²⁶⁵) • 7^(¹⁰ - ⁴⁵)
Step 4: Evaluate the exponents:
= 4^²⁵⁰ • 7^-³⁵
Now, we need to solve the expression:
= (4^²⁵⁰) / (7^³⁵)
This is the final simplified fraction with no remaining exponents.
To begin with, let's apply the properties of integer exponents to generate equivalent expressions for the given expression:
(4³ • 7^-²)^-⁵ / (7⁹ • 4^-¹³)
First, let's simplify the exponents within the parentheses:
(64 • 7^-²)^-⁵ / (7⁹ • 4^-¹³)
Next, we can simplify the exponents of 7 and 4 separately:
(64 • (1/7²))^-⁵ / (7⁹ • (1/4¹³))
Now, let's simplify the expression by distributing the exponent of -5 to all the factors:
(64^-⁵ • (1/7²)^-⁵) / (7⁹^-⁵ • (1/4¹³)^-⁵)
Applying the rule that a negative exponent is equivalent to its reciprocal with a positive exponent:
((1/64⁵ • (7²)⁵) / (7⁹⁵ • (4¹³)⁵)
Expanding the powers:
(1/64⁵ • 7¹⁰) / (7⁴⁵ • 4⁶⁵)
Now we can simplify the expression:
(1/1024 • 7¹⁰) / (7⁴⁵ • 256)
To further simplify the expression, we can divide the numerator by the denominator:
1/1024 • 7¹⁰ / (7⁴⁵ • 256)
Since we are dividing by (7⁴⁵ • 256), we can subtract the exponents of 7:
1/1024 • 7¹⁰⁻⁴⁵ / 256
Simplifying further:
1/1024 • 7⁻³⁵ / 256
To simplify this expression, we can convert 7⁻³⁵ to (1/7)³⁵:
1/1024 • (1/7)³⁵ / 256
Next, we can rearrange the terms:
(1/1024 • 1/256) • (1/7)³⁵
Combining the fractions:
1/262144 • (1/7)³⁵
Finally, we can simplify the expression:
1 / (262144 • 7³⁵)
Therefore, the final simplified expression is 1 / (262144 • 7³⁵), which is a fraction with no remaining exponents.