High School

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26. Simplify [tex]\left(x^3\right)^4[/tex].

a. [tex]x^1[/tex]
b. [tex]x^{12}[/tex]
c. [tex]x^7[/tex]
d. [tex]x^{34}[/tex]

27. Simplify [tex]\left(x^2\right)^7[/tex].

a. [tex]x^{14}[/tex]
b. [tex]x^{12}[/tex]
c. [tex]x^5[/tex]
d. [tex]x^9[/tex]

28. Simplify [tex]\left(3 x^3\right)^4[/tex].

a. [tex]12 x^2[/tex]
b. [tex]81 x^{12}[/tex]
c. [tex]3 x^{12}[/tex]
d. [tex]12 x^{12}[/tex]

29. Simplify [tex]\frac{x^{12}}{x^9}[/tex].

a. 1
b. [tex]x^{21}[/tex]
c. [tex]x^3[/tex]
d. [tex]\frac{1}{x^3}[/tex]

30. Simplify [tex]x^5 \cdot x^5[/tex].

a. [tex]x^0[/tex]
b. [tex]x^{25}[/tex]
c. [tex]x^{10}[/tex]
d. [tex]x[/tex]

31. Simplify [tex]\frac{x^3}{x^2}[/tex].

a. [tex]x[/tex]
b. [tex]x^5[/tex]
c. [tex]x^6[/tex]
d. [tex]x^{-1}[/tex]

32. Simplify [tex]a^{11} \cdot a^{19}[/tex].

a. [tex]a^{30}[/tex]
b. [tex]a^8[/tex]
c. [tex]a^{26}[/tex]
d. [tex]a^{209}[/tex]

33. Simplify [tex]15^{\circ}[/tex].

a. 0
b. 1
c. 15
d. [tex]\frac{1}{15}[/tex]

34. Simplify [tex]\left(-2 x^5\right)^5[/tex].

a. [tex]-32 x^{10}[/tex]
b. [tex]-32 x^{25}[/tex]
c. [tex]-10 x^{10}[/tex]
d. [tex]-10 x^{25}[/tex]

35. Simplify [tex]2^{\circ}[/tex].

a. 1
b. 0
c. 2
d. [tex]\frac{1}{2}[/tex]

36. Simplify [tex]12^{\circ}[/tex].

a. 1
b. 0
c. 12
d. [tex]\frac{1}{12}[/tex]

37. Simplify [tex]3^{-9}[/tex].

a. [tex]\frac{1}{3^9}[/tex]
b. [tex]\frac{1}{3^{-9}}[/tex]
c. [tex]\frac{1}{3}[/tex]
d. -27

Answer :

Sure, I'd be happy to help you understand the solutions step-by-step:

26. Simplify [tex]\((x^3)^4\)[/tex]:

Using the power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:

- Multiply the exponents: [tex]\(3 \times 4 = 12\)[/tex].
- So, the simplified expression is [tex]\(x^{12}\)[/tex].

27. Simplify [tex]\((x^2)^7\)[/tex]:

Again, using the power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:

- Multiply the exponents: [tex]\(2 \times 7 = 14\)[/tex].
- The simplified expression is [tex]\(x^{14}\)[/tex].

28. Simplify [tex]\((3x^3)^4\)[/tex]:

A number and variable inside a power are simplified separately: [tex]\((ab^n)^m = a^m \cdot b^{n \cdot m}\)[/tex].

- The coefficient [tex]\(3\)[/tex] raised to the fourth power: [tex]\(3^4 = 81\)[/tex].
- For the variable part: multiply the exponents [tex]\(3 \times 4 = 12\)[/tex].
- The simplified expression is [tex]\(81x^{12}\)[/tex].

29. Simplify [tex]\(\frac{x^{12}}{x^9}\)[/tex]:

Using the property [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:

- Subtract the exponents: [tex]\(12 - 9 = 3\)[/tex].
- The simplified expression is [tex]\(x^3\)[/tex].

30. Simplify [tex]\(x^5 \cdot x^5\)[/tex]:

Using the property [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]:

- Add the exponents: [tex]\(5 + 5 = 10\)[/tex].
- The simplified expression is [tex]\(x^{10}\)[/tex].

31. Simplify [tex]\(\frac{x^3}{x^2}\)[/tex]:

Using the property [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:

- Subtract the exponents: [tex]\(3 - 2 = 1\)[/tex].
- The simplified expression is [tex]\(x\)[/tex].

32. Simplify [tex]\(a^{11} \cdot a^{19}\)[/tex]:

Using the property [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]:

- Add the exponents: [tex]\(11 + 19 = 30\)[/tex].
- The simplified expression is [tex]\(a^{30}\)[/tex].

33. Simplify [tex]\(15^{0}\)[/tex]:

According to a mathematical rule, any non-zero number raised to the power of zero is [tex]\(1\)[/tex].

- So, [tex]\(15^{0} = 1\)[/tex].

34. Simplify [tex]\((-2x^5)^5\)[/tex]:

A number and variable inside a power are simplified separately:

- The coefficient [tex]\(-2\)[/tex] raised to the fifth power: [tex]\((-2)^5 = -32\)[/tex].
- For the variable part: multiply the exponent [tex]\(5 \times 5 = 25\)[/tex].
- The simplified expression is [tex]\(-32x^{25}\)[/tex].

35. Simplify [tex]\(2^{0}\)[/tex]:

Any non-zero number raised to the power of zero is [tex]\(1\)[/tex].

- So, [tex]\(2^{0} = 1\)[/tex].

36. Simplify [tex]\(12^{0}\)[/tex]:

Following the same rule, any non-zero number raised to the power of zero is [tex]\(1\)[/tex].

- So, [tex]\(12^{0} = 1\)[/tex].

37. Simplify [tex]\(3^{-9}\)[/tex]:

Using the property of negative exponents [tex]\(a^{-n} = \frac{1}{a^n}\)[/tex]:

- This converts to [tex]\(\frac{1}{3^9}\)[/tex].

These are the simplified expressions and explanations for each part of the problem. Let me know if you have any more questions!

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