Simplify The Expression, Leaving The Answer With Positive Exponents:$\[ 3\left(\frac{1}{3}\right)^x = 4 \\]

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Introduction

In mathematics, simplifying expressions with exponents is a crucial skill that helps us solve various problems in algebra, calculus, and other branches of mathematics. In this article, we will focus on simplifying the expression ${ 3\left(\frac{1}{3}\right)^x = 4 }$ and leaving the answer with positive exponents.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, ${ 2^3 }$ means 2×2×22 \times 2 \times 2, which is equal to 88. In the given expression, ${ \left(\frac{1}{3}\right)^x }$, the base is 13\frac{1}{3} and the exponent is xx.

Simplifying the Expression

To simplify the expression, we need to isolate the exponent. We can start by dividing both sides of the equation by 33:

3(13)x=4{ 3\left(\frac{1}{3}\right)^x = 4 }

(13)x=43{ \left(\frac{1}{3}\right)^x = \frac{4}{3} }

Using Exponent Rules

Now, we can use the rule of exponents that states ${ \left(\frac{a}{b}\right)^x = \frac{ax}{bx} }$. In this case, we have:

(13)x=43{ \left(\frac{1}{3}\right)^x = \frac{4}{3} }

1x3x=43{ \frac{1^x}{3^x} = \frac{4}{3} }

Simplifying Further

Since 11 raised to any power is still 11, we can simplify the expression further:

1x3x=43{ \frac{1^x}{3^x} = \frac{4}{3} }

13x=43{ \frac{1}{3^x} = \frac{4}{3} }

Isolating the Exponent

Now, we can isolate the exponent by multiplying both sides of the equation by 3x3^x:

13x=43{ \frac{1}{3^x} = \frac{4}{3} }

1=4⋅3x{ 1 = 4 \cdot 3^x }

Simplifying the Right Side

We can simplify the right side of the equation by using the rule of exponents that states ${ a \cdot a^x = a^{x+1} }$. In this case, we have:

1=4⋅3x{ 1 = 4 \cdot 3^x }

1=4⋅3x+0{ 1 = 4 \cdot 3^{x+0} }

1=4⋅3x⋅30{ 1 = 4 \cdot 3^{x} \cdot 3^0 }

Simplifying Further

Since 303^0 is equal to 11, we can simplify the expression further:

1=4⋅3x⋅30{ 1 = 4 \cdot 3^{x} \cdot 3^0 }

1=4⋅3x⋅1{ 1 = 4 \cdot 3^{x} \cdot 1 }

1=4⋅3x{ 1 = 4 \cdot 3^{x} }

Isolating the Exponent

Now, we can isolate the exponent by dividing both sides of the equation by 44:

1=4⋅3x{ 1 = 4 \cdot 3^{x} }

14=3x{ \frac{1}{4} = 3^{x} }

Using Exponent Rules

Now, we can use the rule of exponents that states ${ \left(\frac{a}{b}\right)^x = \frac{ax}{bx} }$. In this case, we have:

14=3x{ \frac{1}{4} = 3^{x} }

(14)x=3x{ \left(\frac{1}{4}\right)^x = 3^{x} }

Simplifying Further

Since 14\frac{1}{4} can be written as 4−14^{-1}, we can simplify the expression further:

(14)x=3x{ \left(\frac{1}{4}\right)^x = 3^{x} }

(4−1)x=3x{ \left(4^{-1}\right)^x = 3^{x} }

Using Exponent Rules

Now, we can use the rule of exponents that states ${ \left(\frac{a}{b}\right)^x = \frac{ax}{bx} }$. In this case, we have:

(4−1)x=3x{ \left(4^{-1}\right)^x = 3^{x} }

4−x1=3x{ \frac{4^{-x}}{1} = 3^{x} }

Simplifying Further

Since 4−x4^{-x} can be written as 14x\frac{1}{4^x}, we can simplify the expression further:

4−x1=3x{ \frac{4^{-x}}{1} = 3^{x} }

14x=3x{ \frac{1}{4^x} = 3^{x} }

Isolating the Exponent

Now, we can isolate the exponent by multiplying both sides of the equation by 4x4^x:

14x=3x{ \frac{1}{4^x} = 3^{x} }

1=3x⋅4x{ 1 = 3^{x} \cdot 4^x }

Simplifying the Right Side

We can simplify the right side of the equation by using the rule of exponents that states ${ a \cdot a^x = a^{x+1} }$. In this case, we have:

1=3x⋅4x{ 1 = 3^{x} \cdot 4^x }

1=3x⋅4x+0{ 1 = 3^{x} \cdot 4^{x+0} }

1=3x⋅4x⋅40{ 1 = 3^{x} \cdot 4^{x} \cdot 4^0 }

Simplifying Further

Since 404^0 is equal to 11, we can simplify the expression further:

1=3x⋅4x⋅40{ 1 = 3^{x} \cdot 4^{x} \cdot 4^0 }

1=3x⋅4x⋅1{ 1 = 3^{x} \cdot 4^{x} \cdot 1 }

1=3x⋅4x{ 1 = 3^{x} \cdot 4^{x} }

Isolating the Exponent

Now, we can isolate the exponent by taking the logarithm of both sides of the equation:

1=3x⋅4x{ 1 = 3^{x} \cdot 4^{x} }

log⁡(1)=log⁡(3x⋅4x){ \log(1) = \log(3^{x} \cdot 4^{x}) }

Using Logarithm Rules

Now, we can use the rule of logarithms that states ${ \log(a \cdot b) = \log(a) + \log(b) }$. In this case, we have:

log⁡(1)=log⁡(3x⋅4x){ \log(1) = \log(3^{x} \cdot 4^{x}) }

log⁡(1)=log⁡(3x)+log⁡(4x){ \log(1) = \log(3^{x}) + \log(4^{x}) }

Simplifying Further

Since log⁡(1)\log(1) is equal to 00, we can simplify the expression further:

log⁡(1)=log⁡(3x)+log⁡(4x){ \log(1) = \log(3^{x}) + \log(4^{x}) }

0=log⁡(3x)+log⁡(4x){ 0 = \log(3^{x}) + \log(4^{x}) }

Isolating the Exponent

Now, we can isolate the exponent by subtracting log⁡(4x)\log(4^{x}) from both sides of the equation:

0=log⁡(3x)+log⁡(4x){ 0 = \log(3^{x}) + \log(4^{x}) }

0−log⁡(4x)=log⁡(3x){ 0 - \log(4^{x}) = \log(3^{x}) }

Simplifying Further

Since 0−log⁡(4x)0 - \log(4^{x}) is equal to −log⁡(4x)-\log(4^{x}), we can simplify the expression further:

0−log⁡(4x)=log⁡(3x){ 0 - \log(4^{x}) = \log(3^{x}) }

−log⁡(4x)=log⁡(3x){ -\log(4^{x}) = \log(3^{x}) }

Using Logarithm Rules

Now, we can use the rule of logarithms that states ${ \log(a) = \log(b) \implies a = b }$. In this case, we have:

−log⁡(4x)=log⁡(3x){ -\log(4^{x}) = \log(3^{x}) }

4x=3x{ 4^{x} = 3^{x} }

Simplifying Further

Since 4x4^{x} is equal to 3x3^{x}, we can simplify the expression further:

4x=3x{ 4^{x} = 3^{x} }

4x−3x=0{ 4^{x} - 3^{x} = 0 }

Factoring the Difference

We can factor the difference of squares:

4x−3x=0{ 4^{x} - 3^{x} = 0 }

(22)x−(3)x=0{ (2^{2})^{x} - (3)^{x} = 0 }

22x−3x=0{ 2^{2x} - 3^{x} = 0 }

Factoring the Difference

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Introduction

In our previous article, we simplified the expression ${ 3\left(\frac{1}{3}\right)^x = 4 }$ and left the answer with positive exponents. In this article, we will answer some frequently asked questions about simplifying expressions with exponents.

Q: What is the rule for simplifying expressions with exponents?

A: The rule for simplifying expressions with exponents is to isolate the exponent by using the rules of exponents, such as ${ \left(\frac{a}{b}\right)^x = \frac{ax}{bx} }$ and ${ a \cdot a^x = a^{x+1} }$.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you can use the rule ${ a^{-x} = \frac{1}{a^x} }$. For example, ${ 2^{-x} = \frac{1}{2^x} }$.

Q: What is the difference between a positive and negative exponent?

A: A positive exponent means that the base is raised to a power, while a negative exponent means that the base is raised to a power and then taken as a reciprocal. For example, ${ 2^x }$ means 22 raised to the power of xx, while ${ 2^{-x} }$ means 22 raised to the power of −x-x and then taken as a reciprocal.

Q: How do I simplify an expression with a fractional exponent?

A: To simplify an expression with a fractional exponent, you can use the rule ${ a^{m/n} = \sqrt[n]{a^m} }$. For example, ${ 2^{3/4} = \sqrt[4]{2^3} }$.

Q: What is the rule for simplifying expressions with logarithms?

A: The rule for simplifying expressions with logarithms is to use the rule ${ \log(a \cdot b) = \log(a) + \log(b) }$. For example, ${ \log(3^{x} \cdot 4^{x}) = \log(3^{x}) + \log(4^{x}) }$.

Q: How do I simplify an expression with a logarithm and an exponent?

A: To simplify an expression with a logarithm and an exponent, you can use the rule ${ \log(a^x) = x \log(a) }$. For example, ${ \log(3^{x}) = x \log(3) }$.

Q: What is the difference between a linear and exponential function?

A: A linear function is a function that can be written in the form f(x)=mx+bf(x) = mx + b, where mm and bb are constants. An exponential function is a function that can be written in the form f(x)=axf(x) = a^x, where aa is a constant. For example, f(x)=2x+3f(x) = 2x + 3 is a linear function, while f(x)=2xf(x) = 2^x is an exponential function.

Q: How do I graph an exponential function?

A: To graph an exponential function, you can use a graphing calculator or a computer program. You can also use the rule ${ f(x) = a^x }$ to graph the function.

Q: What is the rule for simplifying expressions with absolute value?

A: The rule for simplifying expressions with absolute value is to use the rule ${ |a| = a \text{ if } a \geq 0 \text{ and } |a| = -a \text{ if } a < 0 }$. For example, ${ |3| = 3 \text{ if } 3 \geq 0 \text{ and } |3| = -3 \text{ if } 3 < 0 }$.

Q: How do I simplify an expression with absolute value and an exponent?

A: To simplify an expression with absolute value and an exponent, you can use the rule ${ |a^x| = a^x \text{ if } a^x \geq 0 \text{ and } |a^x| = -a^x \text{ if } a^x < 0 }$. For example, ${ |3^x| = 3^x \text{ if } 3^x \geq 0 \text{ and } |3^x| = -3^x \text{ if } 3^x < 0 }$.

Conclusion

In this article, we have answered some frequently asked questions about simplifying expressions with exponents. We have also discussed the rules for simplifying expressions with logarithms, absolute value, and exponents. We hope that this article has been helpful in understanding the rules for simplifying expressions with exponents.