Apply the properties of integer exponents to generate an equivalent algebraic expression to (4x4y−4)3(x7y3)2.(1 point)
Responses
64x20
Start Fraction 64 over x superscript 20 baseline End Fraction
4x2y18
Start Fraction 4 over x squared y superscript 18 baseline End Fraction
x2y1864
Start Fraction x squared y superscript 18 baseline over 64 End Fraction
64x2y18
The equivalent algebraic expression to (4x4y−4)3(x7y3)2 is 64x2y18.
is
Start Fraction 64 over x superscript 20 End Fraction an equivalent algebraic expression.
Apply the product rule of integer exponents to find the numerical equivalent of 12−5⋅127. Show your work and explain how you used the product rule of integer exponents.(2 points)
To find the numerical equivalent of 12^(-5) * 127, we can apply the product rule of integer exponents.
The product rule states that when multiplying two numbers with the same base, you can add their exponents.
In this case, we have 12^(-5) multiplied by 12^7. Since the bases are the same (12), we can add the exponents (-5 + 7) to get (-5 + 7) = 2.
So, 12^(-5) * 12^7 is equivalent to 12^2.
To calculate the numerical value, we just need to evaluate 12^2, which is 144.
Therefore, the numerical equivalent of 12^(-5) * 127 is 144.
To generate an equivalent expression, we need to apply the properties of integer exponents.
Given the expression (4x^4y^-4)^3(x^7y^3)^2, let's simplify it step by step.
Step 1: Simplify the exponents inside the first parentheses (4x^4y^-4)^3:
- The exponent 3 applies to each term inside the parentheses.
- Multiply the exponents of x: 4 * 3 = 12. So, x^12.
- Multiply the exponents of y: -4 * 3 = -12. So, y^-12.
The expression becomes (4x^12y^-12)(x^7y^3)^2.
Step 2: Simplify the exponents inside the second parentheses (x^7y^3)^2:
- The exponent 2 applies to each term inside the parentheses.
- Multiply the exponents of x: 7 * 2 = 14. So, x^14.
- Multiply the exponents of y: 3 * 2 = 6. So, y^6.
The expression becomes (4x^12y^-12)(x^14y^6).
Step 3: Combine like terms by multiplying the coefficients and adding the exponents with the same bases:
- Multiply the coefficients: 4 * 1 = 4.
- Combine the x terms: x^12 * x^14 = x^(12+14) = x^26.
- Combine the y terms: y^-12 * y^6 = y^(-12+6) = y^-6.
The final equivalent expression is 4x^26y^-6, or in fractional form, Start Fraction 4x^26 over y^6 End Fraction.
To generate an equivalent algebraic expression to (4x^4y^-4)^3(x^7y^3)^2, we need to apply the properties of integer exponents.
First, let's simplify the expression inside the first set of parentheses, (4x^4y^-4)^3. To raise a power to another power, we multiply the exponents:
(4x^4y^-4)^3 = 4^3 * (x^4)^3 * (y^-4)^3
Simplifying further, we have:
64 * x^(4*3) * y^(-4*3) = 64x^12 * y^-12
Next, let's simplify the expression inside the second set of parentheses, (x^7y^3)^2:
(x^7y^3)^2 = (x^7)^2 * (y^3)^2
Again, simplifying further:
x^(7*2) * y^(3*2) = x^14 * y^6
Now, we can multiply the two simplified expressions together:
64x^12 * y^-12 * x^14 * y^6
When multiplying with the same base, we add the exponents:
= 64 * x^(12+14) * y^(-12+6)
= 64 * x^26 * y^-6
Finally, we can rewrite this expression without negative exponents by moving the terms with negative exponents to the denominator:
= 64 * x^26 / y^6
So, the equivalent algebraic expression is 64x^26 / y^6.