Simplify the following expressions involving exponents: a) x⁻³ / y⁻², b) 2xⁿ y4 / 2x^(n−1) y3, c)…

Question

Simplify the following expressions involving exponents: a) x⁻³ / y⁻², b) 2xⁿ y4 / 2x^(n−1) y3, c) ((1/2)⁻²)⁻³, d) (32^(3a+2)) / (8^(a+b))

ScanSolve AI Tutor · Accepted Answer

Here are the simplifications: a) Step 1: Apply the negative exponent rule a⁻ⁿ = (1)/(aⁿ). (x⁻³)/(y⁻²) = (y²)/(x³) The expression is already simplified. (y²)/(x³) b) Step 1: Separate the terms and apply the exponent rule (a^m)/(aⁿ) = a^(m−n). (2xⁿ y⁴)/(2xⁿ⁻¹ y³) = (2/2) ((xⁿ)/(xⁿ⁻¹)) ((y⁴)/(y³)) Step 2: Simplify each part. 1 · xⁿ⁻⁽ⁿ⁻¹⁾ · y⁴⁻³ = xⁿ⁻ⁿ⁺¹ · y¹ Step 3: Combine the terms. x¹ y¹ = xy xy c) Step 1: Simplify the innermost exponent using (a/b)⁻ⁿ = (b/a)ⁿ. ((1/2)⁻²)⁻³ = (2²)⁻³ Step 2: Simplify the term inside the parenthesis. (4)⁻³ Step 3: Apply the negative exponent rule a⁻ⁿ = (1)/(aⁿ). 4⁻³ = (1)/(4³) Step 4: Calculate the final value. (1)/(4 × 4 × 4) = 1/64 1/64 d) Step 1: Express the bases 32 and 8 as powers of 2. 32 = 2⁵; 8 = 2³ Substitute these into the expression. ((2⁵)^(3a+2))/((2³)^(a+b)) Step 2: Apply the exponent rule (a^m)ⁿ = a^(m × n). (2^(5(3a+2)))/(2^(3(a+b))) = (2^(15a+10))/(2^(3a+3b)); Step 3: Apply the exponent rule (a^m)/(aⁿ) = a^(m−n). 2^((15a+10) − (3a+3b)) Step 4: Simplify the exponent. 2^(15a+10−3a−3b) = 2^(12a−3b+10) 2^(12a−3b+10)

Simplify the following expressions involving exponents: a) x⁻³ / y⁻², b) 2xⁿ y4 / 2x^(n-1) y3, c) ((1/2)⁻²)⁻³, d) (32^(3a+2)) / (8^(a+b))

Answers

Step-by-step solution

Part a· 1 step

Apply the negative exponent rule $a^{-n} = \frac{1}{a^n}$ .

Part b· 3 stepsxy

Separate the terms and apply the exponent rule $\frac{a^m}{a^n} = a^{m-n}$ .

Simplify each part.

Combine the terms.

Part c· 4 steps
$\frac{1}{64}$

Simplify the innermost exponent using $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$ .

Simplify the term inside the parenthesis.

Apply the negative exponent rule $a^{-n} = \frac{1}{a^n}$ .

Calculate the final value.

Part d· 4 steps
$2^{12a-3b+10}$

Express the bases 32 and 8 as powers of 2.

Apply the exponent rule $(a^m)^n = a^{m \times n}$ .

Apply the exponent rule $\frac{a^m}{a^n} = a^{m-n}$ .

Simplify the exponent.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Simplify the following expressions involving exponents: a) x⁻³ / y⁻², b) 2xⁿ y4 / 2x^(n-1) y3, c) ((1/2)⁻²)⁻³, d) (32^(3a+2)) / (8^(a+b))

Answers

Step-by-step solution

Part a· 1 step

Apply the negative exponent rule a−n=1ana^{-n} = \frac{1}{a^n}a−n=an1​.

Part b· 3 stepsxy

Separate the terms and apply the exponent rule aman=am−n\frac{a^m}{a^n} = a^{m-n}anam​=am−n.

Simplify each part.

Combine the terms.

Part c· 4 steps164\frac{1}{64}641​

Simplify the innermost exponent using (ab)−n=(ba)n\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n(ba​)−n=(ab​)n.

Simplify the term inside the parenthesis.

Apply the negative exponent rule a−n=1ana^{-n} = \frac{1}{a^n}a−n=an1​.

Calculate the final value.

Part d· 4 steps212a−3b+102^{12a-3b+10}212a−3b+10

Express the bases 32 and 8 as powers of 2.

Apply the exponent rule (am)n=am×n(a^m)^n = a^{m \times n}(am)n=am×n.

Apply the exponent rule aman=am−n\frac{a^m}{a^n} = a^{m-n}anam​=am−n.

Simplify the exponent.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Apply the negative exponent rule $a^{-n} = \frac{1}{a^n}$ .

Separate the terms and apply the exponent rule $\frac{a^m}{a^n} = a^{m-n}$ .

Part c· 4 steps
$\frac{1}{64}$

Simplify the innermost exponent using $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$ .

Apply the negative exponent rule $a^{-n} = \frac{1}{a^n}$ .

Part d· 4 steps
$2^{12a-3b+10}$

Apply the exponent rule $(a^m)^n = a^{m \times n}$ .

Apply the exponent rule $\frac{a^m}{a^n} = a^{m-n}$ .