Exponents Rules

## 1. The 4 Important Exponents Rules

Here is a brief introduction to the four important rules for exponents in algebra:

Product of Powers Rule: When multiplying two terms with the same base, we add the exponents.
Example with numbers: $$2^3 \cdot 2^4 = 2^{3+4} = 2^7$$
Example with variables: $$x^3 \cdot x^4 = x^{3+4} = x^7$$

Quotient of Powers Rule: When dividing two terms with the same base, we subtract the exponents.
Example with numbers: $$\dfrac{3^6}{3^2} = 3^{6-2} = 3^4$$
Example with variables: $$\dfrac{y^6}{y^2} = y^{6-2} = y^4$$

Power of a Power Rule: When you have a power raised to another power, we multiply the exponents.
Example with numbers: $$(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$$
Example with variables: $$(z^3)^4 = z^{3 \cdot 4} = z^{12}$$
Zero Power Rule: Any nonzero number raised to the power of zero is 1.
Example with numbers: $$7^0 = 1$$
Example with variables: $$a^0 = 1$$

Remember, understanding and applying these rules will help you simplify and solve more complex algebraic expressions and equations. Be careful to ensure the bases are the same when applying these rules.

## 2. Negative Exponents

Negative exponents can be understood by using the idea of reciprocals and the properties of exponents.

When a number or variable is raised to a negative exponent, it indicates that the reciprocal of that number or variable should be taken to the positive exponent.

Let's take an example: $$\frac{1}{{4^{-3}}}$$

To simplify this expression, we can rewrite it as the reciprocal with a positive exponent:

$\frac{1}{{4^{-3}}} = \frac{1}{{\frac{1}{{4^3}}}}$

Now, let's simplify the expression within the parentheses:

$$\frac{1}{{4^3}}$$

Here, $$4^3$$ means $$4 \times 4 \times 4$$, which is equal to 64.

Therefore, we have:

$\frac{1}{{\frac{1}{{4^3}}}} = \frac{1}{{\frac{1}{64}}} = 64$

So, $$\frac{1}{{4^{-3}}} = 64$$.

In this case, the negative exponent flips the fraction and changes the sign of the exponent. It allows us to convert a negative exponent into a positive exponent by taking the reciprocal of the base.

## 3. Simplifying Exponent Expressions

Solve This:
Evaluate this expression $$\dfrac{5y^3}{\left(y^{-2}\right)^3}$$

To solve this problem, follow the order of operations and apply the exponent rules.

First, simplify the expression within the parentheses. To raise a power to another power, we multiply the exponents:
$\left(y^{-2}\right)^3 = y^{-2 \times 3} = y^{-6}$

Now, rewrite the original expression. To divide by a negative exponent, rewrite it as a positive exponent:
$\frac{5y^3}{y^{-6}} = \frac{5y^3}{\frac{1}{y^6}}$

Dividing by a fraction is equivalent to multiplying by its reciprocal:
$5y^3 \times \frac{y^6}{1} = 5y^{3 + 6} = 5y^9$

Therefore, the simplified expression is $$5y^9$$.

## 4. Solving Equations with Exponents

Solve the equation: $7^{4a+3} = 7^{a+9}$

To solve the equation, we can equate the exponents:
$4a+3 = a+9 \Rightarrow 3a + 3 = 9$

Solving for $$a$$:
$3a = 6 \Rightarrow a = 2$

Therefore, the solution to the equation is $$a = 2$$.

Solve This:
If $$16^{20} = 2^x$$, then find the value of $$x$$.

We know that $$16 = 2^4$$ because $$2 \times 2 \times 2 \times 2 = 16$$.

Using the property of exponents, we can rewrite the equation as $$(2^4)^{20} = 2^x$$.

Applying the power rule, we multiply the exponents:

$2^{4 \times 20} = 2^x$

Simplifying the exponent on the left side, we have:

$2^{80} = 2^x \Rightarrow 80 = x$

Therefore, in the equation $$16^{20} = 2^x$$, the value of $$x$$ is 80.

Solve This:
If $$4a - b = 2$$, find the value of $$\frac{{81^a}}{{3^b}}$$

We are given that $$4a - b = 2$$.

Let's simplify the expression $$\frac{{81^a}}{{3^b}}$$ using the properties of exponents.

We know that $$81 = 3^4$$ because $$3 \times 3 \times 3 \times 3 = 81$$.

Substituting this into the expression, and using the power of a power rule, we multiply the exponents we have:

$\frac{{(3^4)^a}}{{3^b}} = 3^{4a - b}$

Since we are given that $$4a - b = 2$$, we can substitute it back into the expression:

$3^{4a - b} = 3^2$

Therefore, the value of $$\frac{{81^a}}{{3^b}}$$ when $$4a - b = 2$$ is 9.

## 5. Convert Radicals to Fractional Exponents

Radicals can be represented using fractional exponents, providing a more concise and flexible notation. Here are a few examples:

1. Square Root: The square root of a number can be expressed as the number raised to the power of $$1/2$$. For instance, $$\sqrt{5}$$ is equivalent to $$5^{1/2}$$.

2. Cube Root: The cube root of a number can be represented as the number raised to the power of $$1/3$$. For example, $$\sqrt[3]{5}$$ is equal to $$5^{1/3}$$.

3. Higher Roots: The same concept applies to higher roots. To find the seventh root of $$10^3$$, we write it as $$\sqrt[7]{10^3}$$, which simplifies to $$10^{3/7}$$.

$\sqrt{5} = 5^{1/2}$ $\sqrt[3]{5} = 5^{1/3}$ $\sqrt[7]{10^3} = 10^{3/7}$

Converting radicals to fractional exponents allows for easier calculations and manipulation of expressions involving roots.

Solve This:
If $$\sqrt[7]{10^5} \cdot \sqrt[7]{10^9} = 10^x$$, find $$x$$.

To solve this problem, we'll simplify each radical expression:

$\sqrt[7]{10^5} = (10^5)^{\frac{1}{7}} = 10^{\frac{5}{7}}$
$\sqrt[7]{10^9} = (10^9)^{\frac{1}{7}} = 10^{\frac{9}{7}}$

Now, we can multiply the simplified radicals:

$\sqrt[7]{10^5} \cdot \sqrt[7]{10^9} = 10^{\frac{5}{7}} \cdot 10^{\frac{9}{7}}$

To multiply exponential expressions with the same base, we add the exponents:

$10^{\frac{5}{7}} \cdot 10^{\frac{9}{7}} = 10^{\frac{5}{7} + \frac{9}{7}}$

Combining the fractions:

$10^{\frac{5+9}{7}} = 10^{\frac{14}{7}}$

Simplifying the exponent:

$10^{\frac{14}{7}} = 10^2$

Therefore, the expression $$\sqrt[7]{10^5} \cdot \sqrt[7]{10^9}$$ simplifies to $$10^2$$.

Since this expression is equal to $$10^x$$, we can equate the exponents:

$x = 2$

Hence, the value of $$x$$ in the equation $$10^x$$ is $$2$$.

Solve This:
If $$2\sqrt{x+3} = \sqrt{8x}$$, find $$x$$.

To solve this equation, we will isolate the square root terms and then square both sides:

$2\sqrt{x+3} = \sqrt{8x}$
$(2\sqrt{x+3})^2 = (\sqrt{8x})^2$
$4(x+3) = 8x$

Expanding and simplifying the equation:

$4x + 12 = 8x$

Bringing like terms together:

$4x - 8x = -12$

Combining like terms:

$-4x = -12$

Dividing both sides by $$-4$$ to solve for $$x$$:

$x = 3$

Hence, the solution to the equation $$2\sqrt{x+3} = \sqrt{8x}$$ is $$x = 3$$.