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

Question

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

UpStudy AI · Accepted Answer

Certainly! Let's simplify each of the given expressions step by step.

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**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:** $$ -3x^4 + 8x^2 - 5 $$

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If you have any further questions or need additional explanations, feel free to ask!
Here are the simplified answers for each problem: 1. **Problem 89:** $$ 27\sqrt{2} $$ 2. **Problem 90:** $$ -51\sqrt{y} $$ 3. **Problem 91:** $$ 7^7 $$ 4. **Problem 92:** $$ 10^3 $$ 5. **Problem 93:** $$ 2x^2 + 8x + 11 $$ 6. **Problem 94:** $$ 12x^3 - 4x^2 + 44x $$ 7. **Problem 95:** $$ 2x^2 - 5x - 63 $$ 8. **Problem 96:** $$ -3x^4 + 8x^2 - 5 $$ If you need further assistance or more detailed explanations, please let me know!

Simplifying an Expression In Exercises 89-92,
simplify the expression.

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