Simplify the following expressions: a) (6^(3n+1) × 9^(3n)) / 3^(n+1) b) (5^(2x+1) − 25^x) / (5^(2x+1) + 25^x)

Question

ScanSolve AI Tutor · Accepted Answer

Here's the solution to problem 1. a) Simplify (6³ⁿ⁺¹ · 9³ⁿ)/(3ⁿ⁺¹) Step 1: Express the bases in terms of their prime factors. 6 = 2 × 3; 9 = 3² Step 2: Substitute the prime factors into the expression. ((2 × 3)³ⁿ⁺¹ · (3²)³ⁿ)/(3ⁿ⁺¹) Step 3: Apply the exponent rules (ab)^m = a^m b^m and (a^m)ⁿ = a^(mn). (2³ⁿ⁺¹ · 3³ⁿ⁺¹ · 3^(2 · 3n))/(3ⁿ⁺¹) (2³ⁿ⁺¹ · 3³ⁿ⁺¹ · 3⁶ⁿ)/(3ⁿ⁺¹) Step 4: Combine terms with the same base in the numerator using a^m · aⁿ = a^(m+n). (2³ⁿ⁺¹ · 3⁽³ⁿ⁺¹⁾⁺⁶ⁿ)/(3ⁿ⁺¹) (2³ⁿ⁺¹ · 3⁹ⁿ⁺¹)/(3ⁿ⁺¹) Step 5: Simplify by dividing terms with the same base using (a^m)/(aⁿ) = a^(m−n). 2³ⁿ⁺¹ · 3⁽⁹ⁿ⁺¹⁾⁻⁽ⁿ⁺¹⁾ 2³ⁿ⁺¹ · 3⁹ⁿ⁺¹⁻ⁿ⁻¹ 2³ⁿ⁺¹ · 3⁸ⁿ The simplified expression is 2³ⁿ⁺¹ · 3⁸ⁿ. b) Simplify (5^(2x+1) − 25^x)/(5^(2x+1) + 25^x) Step 1: Express 25 as a power of 5. 25 = 5²; Step 2: Substitute 25 = 5² into the expression. (5^(2x+1) − (5²)^x)/(5^(2x+1) + (5²)^x) (5^(2x+1) − 5^(2x))/(5^(2x+1) + 5^(2x)) Step 3: Factor out the common term 5^(2x) from the numerator and denominator. Recall that 5^(2x+1) = 5^(2x) · 5¹. (5^(2x) · 5 − 5^(2x))/(5^(2x) · 5 + 5^(2x)) (5^(2x)(5 − 1))/(5^(2x)(5 + 1)) Step 4: Cancel out the common factor 5^(2x). (5 − 1)/(5 + 1) 4/6 Step 5: Simplify the fraction. 2/3 The simplified expression is 2/3.

Simplify the following expressions: a) (6^(3n+1) * 9^(3n)) / 3^(n+1) b) (5^(2x+1) - 25^x) / (5^(2x+1) + 25^x)

Answers

Step-by-step solution

Part a· 5 steps

Express the bases in terms of their prime factors.

Substitute the prime factors into the expression.

Apply the exponent rules $(ab)^m = a^m b^m$ and $(a^m)^n = a^{mn}$ .

Combine terms with the same base in the numerator using $a^m \cdot a^n = a^{m+n}$ .

Simplify by dividing terms with the same base using $\frac{a^m}{a^n} = a^{m-n}$ .

Part b· 5 steps
$\frac{2}{3}$

Express 25 as a power of 5.

Substitute $25 = 5^2$ into the expression.

Factor out the common term $5^{2x}$ from the numerator and denominator.

Cancel out the common factor $5^{2x}$ .

Simplify the fraction.

Try one yourself

Still stuck on this one?

More Mathematics Questions