Simplify the following expressions: a) (2x + 3)(x − 4) + (x + 1)2 b) 24x6 / 4x2 c) 2log2 + log5 - log4 d)…

Question

Simplify the following expressions: a) (2x + 3)(x − 4) + (x + 1)2 b) 24x6 / 4x2 c) 2log2 + log5 - log4 d) (4a+8)/2

ScanSolve AI Tutor · Accepted Answer

a) Step 1: Expand the first term (2x + 3)(x − 4). (2x + 3)(x − 4) = 2x(x) + 2x(−4) + 3(x) + 3(−4) = 2x² − 8x + 3x − 12 = 2x² − 5x − 12 Step 2: Expand the second term (x + 1)². (x + 1)² = x² + 2(x)(1) + 1² = x² + 2x + 1 Step 3: Add the expanded terms and combine like terms. (2x² − 5x − 12) + (x² + 2x + 1) = (2x² + x²) + (−5x + 2x) + (−12 + 1) = 3x² − 3x − 11 The simplified expression is: 3x² − 3x − 11 b) Step 1: Divide the coefficients and use the exponent rule (x^m)/(xⁿ) = x^(m−n). (24x⁶)/(4x²) = 24/4 · x⁶⁻² = 6x⁴ The simplified expression is: 6x⁴ c) Step 1: Use the logarithm property nlog a = log aⁿ for the first term. 2log2 = log(2²) = log4 Step 2: Substitute this back into the expression. log4 + log5 − log4 Step 3: Use the logarithm property log a + log b = log (ab) for the first two terms. log(4 × 5) − log4 = log20 − log4; Step 4: Use the logarithm property log a − log b = log (a/b). log(20/4) = log5 The simplified expression is: log5 d) Step 1: Factor out the common term from the numerator. (4a+8)/(2) = (4(a+2))/(2) Step 2: Simplify the fraction. (4(a+2))/(2) = 2(a+2) = 2a+4 The simplified expression is: 2a+4 e) Step 1: Distribute 3x into (x − 2). 3x(x − 2) = 3x² − 6x Step 2: Distribute −2x into (x + 4). −2x(x + 4) = −2x² − 8x Step 3: Combine the results and collect like terms. (3x² − 6x) − (2x² + 8x) = 3x² − 6x − 2x² − 8x = (3x² − 2x²) + (−6x − 8x) = x² − 14x The simplified expression is: x² − 14x f) Step 1: Use the exponent rule a⁻ⁿ = (1)/(aⁿ). 8^(−2/3) = 18^(2/3); Step 2: Use the exponent rule a^(m/n) = (√[n]a)^m. 8^(2/3) = (√[3]8)² Step 3: Calculate the cube root of 8. √[3]8 = 2 Step 4: Square the result. 2² = 4 Step 5: Substitute back into the fraction. 1/4 The simplified expression is: 1/4

Simplify the following expressions: a) (2x + 3)(x - 4) + (x + 1)2 b) 24x6 / 4x2 c) 2log2 + log5 - log4 d) (4a+8)/2

Answers

Step-by-step solution

Part a· 3 steps

Expand the first term $(2x + 3)(x - 4)$ .

Expand the second term $(x + 1)^2$ .

Add the expanded terms and combine like terms.

Part b· 1 step
`\text{6x^4}`

Divide the coefficients and use the exponent rule $\frac{x^m}{x^n} = x^{m-n}$ .

Part c· 4 steps
$\text{log5}$

Use the logarithm property $n\log a = \log a^n$ for the first term.

Substitute this back into the expression.

Use the logarithm property $\log a + \log b = \log (ab)$ for the first two terms.

Use the logarithm property $\log a - \log b = \log \left(\frac{a}{b}\right)$ .

Part d· 2 steps
$\text{2a+4}$

Factor out the common term from the numerator.

Simplify the fraction.

Part e· 3 steps
`\text{x^2 - 14x}`

Distribute $3x$ into $(x - 2)$ .

Distribute $-2x$ into $(x + 4)$ .

Combine the results and collect like terms.

Part f· 5 steps
$\text{}\frac{1}{4}\text{}$

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

Use the exponent rule $a^{\frac{m}{n}} = (\sqrt[n]{a})^m$ .

Calculate the cube root of 8.

Square the result.

Substitute back into the fraction.

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More Mathematics Questions

Simplify the following expressions: a) (2x + 3)(x - 4) + (x + 1)2 b) 24x6 / 4x2 c) 2log2 + log5 - log4 d) (4a+8)/2

Answers

Step-by-step solution

Part a· 3 steps

Expand the first term (2x+3)(x−4)(2x + 3)(x - 4)(2x+3)(x−4).

Expand the second term (x+1)2(x + 1)^2(x+1)2.

Add the expanded terms and combine like terms.

Part b· 1 step\text{6x^4}

Divide the coefficients and use the exponent rule xmxn=xm−n\frac{x^m}{x^n} = x^{m-n}xnxm​=xm−n.

Part c· 4 stepslog5\text{log5}log5

Use the logarithm property nlog⁡a=log⁡ann\log a = \log a^nnloga=logan for the first term.