Simplify the following expressions: b. 8sqrt(2)sqrt(98)−3sqrt(2) and 2. a. (4^x)/(16) = ((8^x)/(8)) x sqrt(2)

Question

ScanSolve AI Tutor · Accepted Answer

b. Simplify without using a calculator (8√(2))/(√(98) − 3√(2)) Step 1: Simplify the radical in the denominator. √(98) = √(49 × 2) = √(49) × √(2) = 7√(2) Step 2: Substitute the simplified radical back into the expression. (8√(2))/(7√(2) − 3√(2)) Step 3: Combine the terms in the denominator. (8√(2))/((7−3)√(2)) = (8√(2))/(4√(2)) Step 4: Simplify the fraction. (8√(2))/(4√(2)) = 8/4 = 2 The final answer is 2. 2. a. Solve for x in the equation below (4^x)/(16) = ((8^x)/(8)) × √(2) Step 1: Express all terms with a base of 2. 4 = 2²; 16 = 2⁴; 8 = 2³; √(2) = 2^(1/2) Step 2: Rewrite the equation using base 2. ((2²)^x)/(2⁴) = (((2³)^x)/(2³)) × 2^(1/2); (2^(2x))/(2⁴) = (2^(3x))/(2³) × 2^(1/2); Step 3: Apply the exponent rules a^m/aⁿ = a^(m−n) and a^m × aⁿ = a^(m+n). 2^(2x−4) = 2^(3x−3) × 2^(1/2); 2^(2x−4) = 2^(3x−3 + 1/2); 2^(2x−4) = 2^(3x − 6/2 + 1/2); 2^(2x−4) = 2^(3x − 5/2) Step 4: Equate the exponents and solve for x. 2x−4 = 3x − 5/2 Multiply the entire equation by 2 to eliminate the fraction: 2(2x−4) = 2(3x − 5/2); 4x−8 = 6x − 5; 4x − 6x = −5 + 8; −2x = 3; x = −3/2 The final answer is −3/2. b. The curve y = ax² + bx + c has a maximum value at (4, 6) and passes through the point (2, −2). Find the values of a, b and c. Step 1: Use the vertex form of a quadratic equation, y = a(x−h)² + k, where (h,k) is the vertex. Given the vertex is (4,6), substitute h=4 and k=6: y = a(x−4)² + 6 Step 2: Substitute the given point (2,−2) into the equation to find the value of a. −2 = a(2−4)² + 6; −2 = a(−2)² + 6; −2 = 4a + 6; −2 − 6 = 4a; −8 = 4a; a = −2; Step 3: Substitute the value of a back into the vertex form and expand to the standard form y = ax² + bx + c. y = −2(x−4)² + 6; y = −2(x² − 8x + 16) + 6; y = −2x² + 16x − 32 + 6; y = −2x² + 16x − 26; Step 4: Identify the values of a, b, and c by comparing with y = ax² + bx + c. a = −2; b = 16; c = −26; The final answer is a=−2, b=16, c=−26.

Simplify the following expressions: b. 8sqrt(2)sqrt(98)-3sqrt(2) and 2. a. (4^x)/(16) = ((8^x)/(8)) x sqrt(2)

Step-by-step solution

Simplify the radical in the denominator.

Substitute the simplified radical back into the expression.

Combine the terms in the denominator.

Simplify the fraction.

Express all terms with a base of 2.

Rewrite the equation using base 2.

Apply the exponent rules $a^m/a^n = a^{m-n}$ and $a^m \times a^n = a^{m+n}$ .

Equate the exponents and solve for $x$ .

Use the vertex form of a quadratic equation, $y = a(x-h)^2 + k$ , where $(h,k)$ is the vertex.

Substitute the given point $(2,-2)$ into the equation to find the value of $a$ .

Substitute the value of $a$ back into the vertex form and expand to the standard form $y = ax^2 + bx + c$ .

Identify the values of $a, b,$ and $c$ by comparing with $y = ax^2 + bx + c$ .

Still stuck on this one?

More Mathematics Questions