tmt said:In my textbook,
it makes the leap from $2(2^{n + 1}) = 2^{n + 2}$ citing the laws of exponents.
I'm not sure which law of exponents it is referring to.
Thanks
Prove It said:$\displaystyle \begin{align*} a^m \cdot a^n = a^{m + n} \end{align*}$, and here you have $\displaystyle \begin{align*} 2^1 \cdot 2^{n + 1} = 2^{1 + n + 1} = 2^{n + 2} \end{align*}$.
The basic rules of exponents include the product rule, quotient rule, power rule, and negative exponent rule. The product rule states that when multiplying two exponential expressions with the same base, you add the exponents. The quotient rule states that when dividing two exponential expressions with the same base, you subtract the exponents. The power rule states that when raising an exponential expression to a power, you multiply the exponents. Lastly, the negative exponent rule states that an exponential expression with a negative exponent can be rewritten as its reciprocal with a positive exponent.
To simplify exponential expressions, you can apply the basic rules of exponents. Start by looking for any like bases and then use the appropriate rule to combine them. If a base appears multiple times, you can apply the power rule to simplify it. You can also use the negative exponent rule to rewrite any negative exponents as positive exponents. Lastly, if there are any parentheses in the expression, you can use the distributive property to expand them and then simplify further.
In an exponential expression, the coefficient is the number that is multiplied by the base. The base, on the other hand, is the number that is raised to a certain power. For example, in the expression 23, the coefficient is 2 and the base is 3.
Yes, you can have a negative exponent in an exponential expression. A negative exponent indicates that the base is in the denominator of the fraction. For example, 2-3 is equivalent to 1/23 or 1/8.
To solve exponential equations, you can use the rules of exponents to simplify the expression and then isolate the variable on one side of the equation. From there, you can either use logarithms or take the logarithm of both sides to solve for the variable. It is important to check your solutions to make sure they are valid in the original equation.