Simplifying an Expression In Exercises 89-92, simplify the expression. \( \begin{array}{ll}\text { 89. } 7 \sqrt{72}-5 \sqrt{18} & \text { 90. }-10 \sqrt{25 y}-\sqrt{y} \\ \text { 91. } 7^{3 / 2} \cdot 7^{11 / 2} & \text { 92. } \frac{10^{17 / 4}}{10^{5 / 4}} \\ \text { Operations with Polynomials In Exercises 93-9 } \\ \text { perform the operation and write the result in standa } \\ \text { form. } \\ \text { 93. }(9 x-4)+\left(2 x^{2}-x+15\right) \\ \text { 94. } 4 x\left(11-x+3 x^{2}\right) \\ \text { 95. }(2 x+9)(x-7) \\ \text { 96. }\left(3 x^{2}-5\right)\left(-x^{2}+1\right)\end{array} \)

Certainly! Let's simplify each of the given expressions step by step. --- **Problem 89.** Simplify \( 7 \sqrt{72} - 5 \sqrt{18} \) **Solution:** 1. **Factor the radicals to simplify:** \[ \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2} \] \[ \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} \] 2. **Substitute back into the expression:** \[ 7 \times 6\sqrt{2} - 5 \times 3\sqrt{2} = 42\sqrt{2} - 15\sqrt{2} \] 3. **Combine like terms:** \[ (42 - 15)\sqrt{2} = 27\sqrt{2} \] **Answer:** \( 27\sqrt{2} \) --- **Problem 90.** Simplify \( -10 \sqrt{25 y} - \sqrt{y} \) **Solution:** 1. **Simplify the radicals:** \[ \sqrt{25y} = \sqrt{25} \times \sqrt{y} = 5\sqrt{y} \] 2. **Substitute back into the expression:** \[ -10 \times 5\sqrt{y} - \sqrt{y} = -50\sqrt{y} - \sqrt{y} \] 3. **Combine like terms:** \[ (-50 - 1)\sqrt{y} = -51\sqrt{y} \] **Answer:** \( -51\sqrt{y} \) --- **Problem 91.** Simplify \( 7^{3/2} \cdot 7^{11/2} \) **Solution:** 1. **Use the property of exponents:** \( a^m \cdot a^n = a^{m+n} \) \[ 7^{3/2} \cdot 7^{11/2} = 7^{(3/2 + 11/2)} = 7^{14/2} = 7^{7} \] **Answer:** \( 7^7 \) --- **Problem 92.** Simplify \( \frac{10^{17/4}}{10^{5/4}} \) **Solution:** 1. **Use the property of exponents:** \( \frac{a^m}{a^n} = a^{m-n} \) \[ \frac{10^{17/4}}{10^{5/4}} = 10^{(17/4 - 5/4)} = 10^{12/4} = 10^3 \] **Answer:** \( 10^3 \) --- **Problem 93.** Perform the operation and write the result in standard form: \( (9x - 4) + (2x^{2} - x + 15) \) **Solution:** 1. **Remove parentheses:** \[ 9x - 4 + 2x^2 - x + 15 \] 2. **Combine like terms:** - \( 2x^2 \) (only quadratic term) - \( 9x - x = 8x \) (linear terms) - \( -4 + 15 = 11 \) (constant terms) 3. **Arrange in standard form (descending powers of \( x \)):** \[ 2x^2 + 8x + 11 \] **Answer:** \( 2x^2 + 8x + 11 \) --- **Problem 94.** Perform the operation and write the result in standard form: \( 4x(11 - x + 3x^{2}) \) **Solution:** 1. **Distribute \( 4x \) to each term inside the parentheses:** \[ 4x \times 11 + 4x \times (-x) + 4x \times 3x^2 = 44x - 4x^2 + 12x^3 \] 2. **Arrange in standard form (descending powers of \( x \)):** \[ 12x^3 - 4x^2 + 44x \] **Answer:** \( 12x^3 - 4x^2 + 44x \) --- **Problem 95.** Perform the operation and write the result in standard form: \( (2x + 9)(x - 7) \) **Solution:** 1. **Use the FOIL method (First, Outer, Inner, Last):** \[ (2x)(x) + (2x)(-7) + (9)(x) + (9)(-7) = 2x^2 - 14x + 9x - 63 \] 2. **Combine like terms:** \[ 2x^2 - 5x - 63 \] **Answer:** \( 2x^2 - 5x - 63 \) --- **Problem 96.** Perform the operation and write the result in standard form: \( (3x^{2} - 5)(-x^{2} + 1) \) **Solution:** 1. **Distribute each term in the first polynomial to each term in the second polynomial:** \[ 3x^2 \times (-x^2) + 3x^2 \times 1 + (-5) \times (-x^2) + (-5) \times 1 \] \[ = -3x^4 + 3x^2 + 5x^2 - 5 \] 2. **Combine like terms:** \[ -3x^4 + (3x^2 + 5x^2) - 5 = -3x^4 + 8x^2 - 5 \] 3. **Arrange in standard form (descending powers of \( x \)):** \[ -3x^4 + 8x^2 - 5 \] **Answer:** \( -3x^4 + 8x^2 - 5 \) --- If you have any further questions or need additional explanations, feel free to ask!