Simplify each expression below. Assume that the denominator in part (b) is not equal to zero.
a. (x^3)(x^−2)
b. 4^−1
c. (4x^2)^3
d. Y^5/Y^-2
a. When multiplying/dividing, exponents are added/subtracted respectively.
b .4
c. 64x^6
d. When multiplying/dividing, exponents are added/subtracted respectively.
To simplify each expression, we can use the rules of exponents. Here's how:
a. (x^3)(x^−2):
To simplify this expression, we can use the rule that says when we multiply two exponential expressions with the same base, we add their exponents. Therefore, we can rewrite this expression as x^(3 + -2). Simplifying the exponents gives us x^1, which is just x.
b. 4^−1:
To simplify this expression, we can use the rule that says when we have a negative exponent, it means taking the reciprocal of the base with the positive exponent. Therefore, 4^−1 is the same as 1/4^1, which is equal to 1/4.
c. (4x^2)^3:
To simplify this expression, we can use the rule that says when we have an exponent outside parentheses, it applies to every term inside the parentheses. Therefore, we can rewrite this expression as 4^3 * (x^2)^3. Simplifying the exponent inside the parentheses gives us 4^3 * x^(2 * 3), which is equal to 64x^6.
d. Y^5/Y^−2:
To simplify this expression, we can use the rule that says when we divide two exponential expressions with the same base, we subtract their exponents. Therefore, we can rewrite this expression as Y^(5 - -2). Simplifying the exponents gives us Y^(5 + 2), which is equal to Y^7.
So, the simplified expressions are:
a. x
b. 1/4
c. 64x^6
d. Y^7