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Of course! Let's go through each expression step-by-step to solve them.
### Expression (d): [tex]\((-6 d)^t \times (3 d)^2\)[/tex]
1. Break down and simplify the expression.
2. Evaluate the exponentiation and multiplication separately.
3. Combine the simplified parts.
From the expression:
[tex]\[ (-6 d)^t \times (3 d)^2 \][/tex]
First, expand [tex]\((3 d)^2\)[/tex]:
[tex]\[ (3 d)^2 = 9 d^2 \][/tex]
Now, multiply this result by [tex]\((-6 d)^t\)[/tex]:
[tex]\[ (-6 d)^t \times 9 d^2 = 9 d^2 \times (-6 d)^t \][/tex]
Thus, the simplified form is:
[tex]\[ 9 d^2 (-6 d)^t \][/tex]
### Expression (3): [tex]\(6 z^2 + 4 z^5\)[/tex]
We combine like terms (if any) in this polynomial:
[tex]\[ 6 z^2 + 4 z^5 \][/tex]
Since [tex]\(z^2\)[/tex] and [tex]\(z^5\)[/tex] are different terms, this polynomial is already in its simplest form.
### Expression (a): [tex]\(y^5 x y^5 \times y^5\)[/tex]
First, combine the exponential terms with the same base [tex]\(y\)[/tex]:
[tex]\[ y^5 \times y^5 \times y^5 = y^{5+5+5} = y^{15} \][/tex]
Now, include the remaining [tex]\(x\)[/tex]:
[tex]\[ xy^{15} \][/tex]
### Expression (3): [tex]\(z^4 \times z^5\)[/tex]
Here, combine the like terms by adding the exponents:
[tex]\[ z^4 \times z^5 = z^{4+5} = z^9 \][/tex]
### Equation (4): [tex]\(6 x^2 = 61\)[/tex]
To solve for [tex]\(x\)[/tex], divide both sides by 6:
[tex]\[ x^2 = \frac{61}{6} \][/tex]
Take the square root of both sides:
[tex]\[ x = \pm \sqrt{\frac{61}{6}} \][/tex]
This can be further simplified to:
[tex]\[ x = \pm \frac{\sqrt{366}}{6} \][/tex]
### Expression (5): [tex]\(3 x^{-1}\)[/tex]
Recall that [tex]\(x^{-1}\)[/tex] is the same as [tex]\(\frac{1}{x}\)[/tex]:
[tex]\[ 3 x^{-1} = \frac{3}{x} \][/tex]
### Expression (6): [tex]\((36 + ds) 1\)[/tex]
Multiply the terms:
[tex]\[ (36 + ds) \times 1 = 36 + ds \][/tex]
### Expression (i): [tex]\(\left(6 x y^3\right)^2\)[/tex]
Apply the power of a product rule [tex]\((ab)^2 = a^2 b^2\)[/tex]:
[tex]\[ \left(6 x y^3\right)^2 = (6)^2 (x)^2 (y^3)^2 = 36 x^2 y^6 \][/tex]
### Expression (8): [tex]\(x^3 \div x\)[/tex]
Simplify the division by subtracting the exponents:
[tex]\[ x^3 \div x = x^{3-1} = x^2 \][/tex]
### Final Results
Let's collect all the solutions:
1. [tex]\((-6 d)^t \times (3 d)^2\)[/tex] simplifies to [tex]\(9 d^2 (-6 d)^t\)[/tex]
2. [tex]\(6 z^2 + 4 z^5\)[/tex] remains as [tex]\(4 z^5 + 6 z^2\)[/tex]
3. [tex]\(y^5 x y^5 \times y^5\)[/tex] simplifies to [tex]\(x y^{15}\)[/tex]
4. [tex]\(z^4 \times z^5\)[/tex] simplifies to [tex]\(z^9\)[/tex]
5. The solution to [tex]\(6 x^2 = 61\)[/tex] is [tex]\(x = \pm \frac{\sqrt{366}}{6}\)[/tex]
6. [tex]\(3 x^{-1}\)[/tex] simplifies to [tex]\(\frac{3}{x}\)[/tex]
7. [tex]\((36 + ds) 1\)[/tex] simplifies to [tex]\(36 + ds\)[/tex]
8. [tex]\(\left(6 x y^3\right)^2\)[/tex] simplifies to [tex]\(36 x^2 y^6\)[/tex]
9. [tex]\(x^3 \div x\)[/tex] simplifies to [tex]\(x^2\)[/tex]
These are all the simplified forms or solutions to the given expressions and equation.
### Expression (d): [tex]\((-6 d)^t \times (3 d)^2\)[/tex]
1. Break down and simplify the expression.
2. Evaluate the exponentiation and multiplication separately.
3. Combine the simplified parts.
From the expression:
[tex]\[ (-6 d)^t \times (3 d)^2 \][/tex]
First, expand [tex]\((3 d)^2\)[/tex]:
[tex]\[ (3 d)^2 = 9 d^2 \][/tex]
Now, multiply this result by [tex]\((-6 d)^t\)[/tex]:
[tex]\[ (-6 d)^t \times 9 d^2 = 9 d^2 \times (-6 d)^t \][/tex]
Thus, the simplified form is:
[tex]\[ 9 d^2 (-6 d)^t \][/tex]
### Expression (3): [tex]\(6 z^2 + 4 z^5\)[/tex]
We combine like terms (if any) in this polynomial:
[tex]\[ 6 z^2 + 4 z^5 \][/tex]
Since [tex]\(z^2\)[/tex] and [tex]\(z^5\)[/tex] are different terms, this polynomial is already in its simplest form.
### Expression (a): [tex]\(y^5 x y^5 \times y^5\)[/tex]
First, combine the exponential terms with the same base [tex]\(y\)[/tex]:
[tex]\[ y^5 \times y^5 \times y^5 = y^{5+5+5} = y^{15} \][/tex]
Now, include the remaining [tex]\(x\)[/tex]:
[tex]\[ xy^{15} \][/tex]
### Expression (3): [tex]\(z^4 \times z^5\)[/tex]
Here, combine the like terms by adding the exponents:
[tex]\[ z^4 \times z^5 = z^{4+5} = z^9 \][/tex]
### Equation (4): [tex]\(6 x^2 = 61\)[/tex]
To solve for [tex]\(x\)[/tex], divide both sides by 6:
[tex]\[ x^2 = \frac{61}{6} \][/tex]
Take the square root of both sides:
[tex]\[ x = \pm \sqrt{\frac{61}{6}} \][/tex]
This can be further simplified to:
[tex]\[ x = \pm \frac{\sqrt{366}}{6} \][/tex]
### Expression (5): [tex]\(3 x^{-1}\)[/tex]
Recall that [tex]\(x^{-1}\)[/tex] is the same as [tex]\(\frac{1}{x}\)[/tex]:
[tex]\[ 3 x^{-1} = \frac{3}{x} \][/tex]
### Expression (6): [tex]\((36 + ds) 1\)[/tex]
Multiply the terms:
[tex]\[ (36 + ds) \times 1 = 36 + ds \][/tex]
### Expression (i): [tex]\(\left(6 x y^3\right)^2\)[/tex]
Apply the power of a product rule [tex]\((ab)^2 = a^2 b^2\)[/tex]:
[tex]\[ \left(6 x y^3\right)^2 = (6)^2 (x)^2 (y^3)^2 = 36 x^2 y^6 \][/tex]
### Expression (8): [tex]\(x^3 \div x\)[/tex]
Simplify the division by subtracting the exponents:
[tex]\[ x^3 \div x = x^{3-1} = x^2 \][/tex]
### Final Results
Let's collect all the solutions:
1. [tex]\((-6 d)^t \times (3 d)^2\)[/tex] simplifies to [tex]\(9 d^2 (-6 d)^t\)[/tex]
2. [tex]\(6 z^2 + 4 z^5\)[/tex] remains as [tex]\(4 z^5 + 6 z^2\)[/tex]
3. [tex]\(y^5 x y^5 \times y^5\)[/tex] simplifies to [tex]\(x y^{15}\)[/tex]
4. [tex]\(z^4 \times z^5\)[/tex] simplifies to [tex]\(z^9\)[/tex]
5. The solution to [tex]\(6 x^2 = 61\)[/tex] is [tex]\(x = \pm \frac{\sqrt{366}}{6}\)[/tex]
6. [tex]\(3 x^{-1}\)[/tex] simplifies to [tex]\(\frac{3}{x}\)[/tex]
7. [tex]\((36 + ds) 1\)[/tex] simplifies to [tex]\(36 + ds\)[/tex]
8. [tex]\(\left(6 x y^3\right)^2\)[/tex] simplifies to [tex]\(36 x^2 y^6\)[/tex]
9. [tex]\(x^3 \div x\)[/tex] simplifies to [tex]\(x^2\)[/tex]
These are all the simplified forms or solutions to the given expressions and equation.
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