Rewrite the numerator using the exponent rule a^m+n = a^m * aⁿ.

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Question 1.1.1 Step 1: Rewrite the numerator using the exponent rule a^(m+n) = a^m · aⁿ. (2^(x+5) + 2^x)/(2^x) = (2^x · 2⁵ + 2^x)/(2^x) Step 2: Factor out the common term 2^x from the numerator. (2^x(2⁵ + 1))/(2^x) Step 3: Cancel out the common term 2^x from the numerator and denominator. 2⁵ + 1 Step 4: Calculate the value. 32 + 1 = 33 Question 1.1.2 Step 1: Rewrite the terms in the numerator to have a common factor. (2(2^(2x)) − 7 · 2^(2x−1))/(2^(2x−3)) = (2^(2x+1) − 7 · 2^(2x−1))/(2^(2x−3)) Step 2: Express 2^(2x+1) as 2^(2x−1) · 2². (2^(2x−1) · 2² − 7 · 2^(2x−1))/(2^(2x−3)) Step 3: Factor out 2^(2x−1) from the numerator. (2^(2x−1)(2² − 7))/(2^(2x−3)) = (2^(2x−1)(4 − 7))/(2^(2x−3)) Step 4: Simplify the expression in the parenthesis. (2^(2x−1)(−3))/(2^(2x−3)) Step 5: Use the exponent rule (a^m)/(aⁿ) = a^(m−n). −3 · 2^((2x−1) − (2x−3)) Step 6: Simplify the exponent. −3 · 2^(2x−1−2x+3) = −3 · 2² Step 7: Calculate the final value. −3 · 4 = −12 Question 1.2.1 Step 1: Rearrange the equation into standard quadratic form ax² + bx + c = 0. x² − 24 = 5x; x² − 5x − 24 = 0 Step 2: Factor the quadratic expression. Find two numbers that multiply to −24 and add to −5 (these are −8 and 3). (x−8)(x+3) = 0 Step 3: Set each factor equal to zero and solve for x. x−8 = 0 ⇒ x = 8; x+3 = 0 ⇒ x = −3; The solutions are x=8 or x=−3. Question 1.2.2 Step 1: Square both sides of the equation to eliminate the square root. √(2x−1) = 3; (√(2x−1))² = 3²; 2x−1 = 9 Step 2: Add 1 to both sides of the equation. 2x = 9 + 1; 2x = 10 Step 3: Divide by 2 to solve for x. x = 10/2; x = 5

Rewrite the numerator using the exponent rule a^m+n = a^m * aⁿ.

Step-by-step solution

Question 1.1.1

Rewrite the numerator using the exponent rule $a^{m+n} = a^m \cdot a^n$ .

Factor out the common term $2^x$ from the numerator.

Cancel out the common term $2^x$ from the numerator and denominator.

Calculate the value.

Question 1.1.2

Rewrite the terms in the numerator to have a common factor.

Express $2^{2x+1}$ as $2^{2x-1} \cdot 2^2$ .

Factor out $2^{2x-1}$ from the numerator.

Simplify the expression in the parenthesis.

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

Simplify the exponent.

Calculate the final value.

Question 1.2.1

Rearrange the equation into standard quadratic form $ax^2 + bx + c = 0$ .

Factor the quadratic expression.

Set each factor equal to zero and solve for $x$ .

Question 1.2.2

Square both sides of the equation to eliminate the square root.

Add 1 to both sides of the equation.

Divide by 2 to solve for $x$ .

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More Mathematics Questions

Rewrite the numerator using the exponent rule a^m+n = a^m * aⁿ.

Step-by-step solution

Question 1.1.1

Rewrite the numerator using the exponent rule am+n=am⋅ana^{m+n} = a^m \cdot a^nam+n=am⋅an.

Factor out the common term 2x2^x2x from the numerator.

Cancel out the common term 2x2^x2x from the numerator and denominator.

Calculate the value.

Question 1.1.2

Rewrite the terms in the numerator to have a common factor.

Express 22x+12^{2x+1}22x+1 as 22x−1⋅222^{2x-1} \cdot 2^222x−1⋅22.

Factor out 22x−12^{2x-1}22x−1 from the numerator.

Simplify the expression in the parenthesis.

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

Simplify the exponent.

Calculate the final value.

Question 1.2.1

Rearrange the equation into standard quadratic form ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0.

Factor the quadratic expression.

Set each factor equal to zero and solve for xxx.

Question 1.2.2

Square both sides of the equation to eliminate the square root.

Add 1 to both sides of the equation.

Divide by 2 to solve for xxx.

Still stuck on this one?

More Mathematics Questions

Rewrite the numerator using the exponent rule $a^{m+n} = a^m \cdot a^n$ .

Factor out the common term $2^x$ from the numerator.

Cancel out the common term $2^x$ from the numerator and denominator.

Express $2^{2x+1}$ as $2^{2x-1} \cdot 2^2$ .

Factor out $2^{2x-1}$ from the numerator.

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

Rearrange the equation into standard quadratic form $ax^2 + bx + c = 0$ .

Set each factor equal to zero and solve for $x$ .

Divide by 2 to solve for $x$ .