Exponent multiplication If the exponents have distinct bases, their powers cannot be added. If the exponents' bases have coefficients, multiply the coefficients together. Even though the exponents have distinct bases, the coefficients can be multiplied together. For example, suppose that $3^4 + 7^4 = 37$. The bases of the exponents are $3$ and $7$, so they cannot be added. However, the coefficient of $x^{12}$ in both terms is equal to 1, so we can multiply them together to get $37$.

Exponentiation Note that when multiplying **two positive integers**, if one number is a power of another then the result is a power of **the original number**. This property does not hold for adding or subtracting powers of integers. For example, $3^{10} Eq 10^{3}$.

In general, if $a$ and $b$ are positive integers and $b > 1$, then there exists **a unique positive integer** $n$ such that $b^n = a$. We call this number $\text{lcm}(a, b)$ and write it as $a\vert b \vert n$.

- How do you add exponents with different bases and powers?
- How do you add indices with different bases?
- What happens when you add exponents to the same base?
- When the bases are the same and, in addition, the powers are?
- What powers have the same basis?
- Can you multiply exponents with different bases and powers?

Exponents multiplied using various bases have First, add the bases together. Then, multiply by the exponent. Keep the two exponents the same instead of adding them together. Finally, divide by the number base used for the index.

For example, if you had a list of numbers and wanted to create an index for faster searching, you could do so like this: 23, 42, 64. The first number in the index creates an index of length 1. The second number in the index creates an index of length 2, and so on. Doing so would produce the result 21. Searching this index would return true when searching for 64 because that is the only number that matches.

This works because integers are base 10, meaning that they use digits from 0-9 to represent each number. Exponents are raised to the power of base 10, which means that 100 equals 10, 1000 equals 102, etc. Therefore, multiplying integers uses their corresponding positions in a string to create a new integer. In the example above, this would result in 23 * 10^1 + 42 * 10^2 + 64 * 10^3 = 21. Division by **the number base** used for the index (in **this case**, 10) then gives us the result 21.

Keep the same base and add the powers together to multiply phrases with the same base. Raise the product of the bases to the power to multiply phrases with distinct bases but the same power. If the exponents have coefficients associated with their bases, multiply the coefficients together. This method can be used to combine multiple powers of the same number.

When two exponents with the same base are multiplied, their powers sum. When we split two exponents with the same base, their powers will add. For example:

7^(3+4) = 7 * 7 * 7 = 49*7 = 343

3^(2+4) = 3 * 3 * 3 = 27*3 = 81

Thus, 7^3 and 3^5 are both prime numbers.

Multiplication. If two powers have the same foundation, they can be multiplied. We add the exponents of two powers when we multiply them. For example, if you want to find out what four times seven equals, you would write 4 * 7 = 28.

Division. If one power can be divided by another, they can be divided again. You divide both the numerator and denominator of a fraction by **the same power**. So, if you wanted to know what four over one equals, you would write 4 รท 1 = 4.

Root. Roots are special cases where you take a power back to its base. So, taking **the square root** of a number is the same as raising it to the 0.5 power. Squaring a number to get **its cube root** is the same as raising it to the -0.16667 power. And so on for **other roots**.

Exponentiation. Exponents are like powers with a higher degree. So, raising something to the 100th power is the same as raising it to the power 100. Therefore, exponentiation is multiplication with respect to the base raised to the exponent.

When multiplying two integers or variables with the same base, the exponents are simply added. There is no rule for simplifying the operation of multiplying equations with **diverse bases** and exponents. You can multiply numbers with different bases as long as they are expressed in the same unit system. For example, you can multiply meters by centimeters per second because they are expressed in meters per second.

You should be aware that reducing fractions to **lowest terms** is not always possible. For example, there are no whole number solutions for $7^{1/3}$. However, this does not mean that it is impossible to multiply exponents with different bases and powers. It may be possible to divide one exponent by another, thus dividing **the corresponding bases**. For example, if $a^3$ is divided by $b$, then the result must be a multiple of $b$ because $a$ was assumed to be non-zero. This means that at least one of $a,b$ must be divisible by $b$.

In general, if $\gcd(a,b) = 1$, then $a^{\alpha} b^{\beta} = c \;\;(\mathrm{where}\;c \in \mathbb{Z})$ if and only if $\alpha + \beta < 0$ or $\alpha \beta > 0$.