Determine The Value Of Variables \[$a\$\], \[$b\$\], And \[$c\$\] That Make Each Equation True.1. What Is The Value Of \[$a\$\] In This Equation? $\[ \begin{array}{l} \left(x^a\right)^6=\frac{1}{x^{30}} \\

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**Determine the Value of Variables [a], [b], and [c] that Make Each Equation True**

Equation 1: Finding the Value of [a]

What is the Value of [a] in this Equation?

(xa)6=1x30{ \begin{array}{l} \left(x^a\right)^6=\frac{1}{x^{30}} \\ \end{array} }

To find the value of [a], we need to simplify the left-hand side of the equation and equate the exponents.

Step 1: Simplify the Left-Hand Side

Using the power rule of exponents, we can rewrite the left-hand side as:

(xa)6=x6a{ \begin{array}{l} \left(x^a\right)^6=x^{6a} \\ \end{array} }

Step 2: Equate the Exponents

Now, we can equate the exponents on both sides of the equation:

6a=−30{ \begin{array}{l} 6a=-30 \\ \end{array} }

Step 3: Solve for [a]

To solve for [a], we can divide both sides of the equation by 6:

a=−5{ \begin{array}{l} a=-5 \\ \end{array} }

Therefore, the value of [a] is -5.

Q&A

Q: What is the value of [a] in the equation ${ \begin{array}{l} \left(xa\right)6=\frac{1}{x^{30}} \ \end{array} }$?

A: The value of [a] is -5.

Q: How do we simplify the left-hand side of the equation?

A: We use the power rule of exponents to rewrite the left-hand side as ${ \begin{array}{l} \left(xa\right)6=x^{6a} \ \end{array} }$.

Q: How do we equate the exponents on both sides of the equation?

A: We set the exponents equal to each other: 6a=−306a=-30.

Q: How do we solve for [a]?

A: We divide both sides of the equation by 6: a=−5a=-5.

Equation 2: Finding the Value of [b]

What is the Value of [b] in this Equation?

xbx3=x2b−3{ \begin{array}{l} \frac{x^b}{x^3}=x^{2b-3} \\ \end{array} }

To find the value of [b], we need to simplify the left-hand side of the equation and equate the exponents.

Step 1: Simplify the Left-Hand Side

Using the quotient rule of exponents, we can rewrite the left-hand side as:

xb⋅x−3=xb−3{ \begin{array}{l} x^b \cdot x^{-3}=x^{b-3} \\ \end{array} }

Step 2: Equate the Exponents

Now, we can equate the exponents on both sides of the equation:

b−3=2b−3{ \begin{array}{l} b-3=2b-3 \\ \end{array} }

Step 3: Solve for [b]

To solve for [b], we can add 3 to both sides of the equation and then subtract 2b from both sides:

b=0{ \begin{array}{l} b=0 \\ \end{array} }

Therefore, the value of [b] is 0.

Q&A

Q: What is the value of [b] in the equation ${ \begin{array}{l} \frac{xb}{x3}=x^{2b-3} \ \end{array} }$?

A: The value of [b] is 0.

Q: How do we simplify the left-hand side of the equation?

A: We use the quotient rule of exponents to rewrite the left-hand side as ${ \begin{array}{l} x^b \cdot x{-3}=x{b-3} \ \end{array} }$.

Q: How do we equate the exponents on both sides of the equation?

A: We set the exponents equal to each other: b−3=2b−3b-3=2b-3.

Q: How do we solve for [b]?

A: We add 3 to both sides of the equation and then subtract 2b from both sides: b=0b=0.

Equation 3: Finding the Value of [c]

What is the Value of [c] in this Equation?

(xc)4â‹…x6=x3c+6{ \begin{array}{l} \left(x^c\right)^4 \cdot x^6=x^{3c+6} \\ \end{array} }

To find the value of [c], we need to simplify the left-hand side of the equation and equate the exponents.

Step 1: Simplify the Left-Hand Side

Using the power rule of exponents, we can rewrite the left-hand side as:

x4câ‹…x6=x4c+6{ \begin{array}{l} x^{4c} \cdot x^6=x^{4c+6} \\ \end{array} }

Step 2: Equate the Exponents

Now, we can equate the exponents on both sides of the equation:

4c+6=3c+6{ \begin{array}{l} 4c+6=3c+6 \\ \end{array} }

Step 3: Solve for [c]

To solve for [c], we can subtract 3c from both sides of the equation and then subtract 6 from both sides:

c=0{ \begin{array}{l} c=0 \\ \end{array} }

Therefore, the value of [c] is 0.

Q&A

Q: What is the value of [c] in the equation ${ \begin{array}{l} \left(xc\right)4 \cdot x6=x{3c+6} \ \end{array} }$?

A: The value of [c] is 0.

Q: How do we simplify the left-hand side of the equation?

A: We use the power rule of exponents to rewrite the left-hand side as ${ \begin{array}{l} x^{4c} \cdot x6=x{4c+6} \ \end{array} }$.

Q: How do we equate the exponents on both sides of the equation?

A: We set the exponents equal to each other: 4c+6=3c+64c+6=3c+6.

Q: How do we solve for [c]?

A: We subtract 3c from both sides of the equation and then subtract 6 from both sides: c=0c=0.