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!
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!
