Algebra Warmups - Exponents
Algebra is not a list of techniques to memorize. It is a symbolic language that you can experiment with and use however you want as a problem-solving tool. You can solve problems beyond the ability of any calculator if you approach them strategically (and strategy can also save you a lot of time!)
\[\huge \text{Simplify:}\]
\[ \huge \frac{ \color{green}2 ^ \color{purple}{3} \times \color{green}2 ^ \color{purple}{7} } { \color{green}2 ^ {\color{purple}{6}} \times \color{green}2 ^ {\color{purple}{3}}}. \]
What's really going on in this expression?
\(\color{green}2 ^ \color{purple}{3}\) means "multiply three 2's together: \(2 \times 2 \times 2\)." Breaking down each exponent this way, our expression now appears:
\[\frac{( \color{green}2 \times \color{green}2 \times \color{green}2)( \color{green}2 \times \color{green}2 \times \color{green}2 \times \color{green}2 \times \color{green}2 \times \color{green}2 \times \color{green}2)}{( \color{green}2 \times \color{green}2 \times \color{green}2 \times \color{green}2 \times \color{green}2 \times \color{green}2)( \color{green}2 \times \color{green}2 \times \color{green}2)}\]
In total, there are 10 2's in the numerator and 9 total in the denominator of the fraction. We can then simplify further, seeing that nine of the 10 2's in the numerator are cancelled out by 2's in the denominator. So the final answer is simply \(\fbox{2}\).
The above explanation can be your intuition for how the rules of exponents might be applied to solve this problem:
Using the rules of exponents: \(2^{b+c} = 2^b \times 2^c \), the expression is equal to
\[ \frac{ 2 ^ { 3 + 7 } } { 2 ^ { 6 + 3 } } = \frac{ 2 ^ { 10 } } { 2^ { 9} } . \]
Then, again using the rules of exponents \( \dfrac{ 2 ^ a } { 2 ^b } = 2 ^ { a - b } \), we obtain
\[ \frac{ 2 ^ { 10 } } { 2^ { 9} } = 2 ^ 1 = \boxed 2 . \]
You've probably sat through at least one terrible Algebra class where you were given lists of rules and procedures to memorize and apply in sequence. We want your use of this site to be the opposite kind of experience: one where you have many "AH HA!" moments of discovery and accomplishment, when some concept or strategy finally crystallizes in your mind.
\[\Large \begin{array} {c c c } & \color{blue}A & \color{red}B & \color{orange}C \\ \times& & & 8 \\ \hline \color{blue}A & \color{orange}C & \color{orange}C & \color{orange}C \\ \end{array} \]
In the cryptogram above, \(A\), \(B\), and \(C\) are distinct digits. Find the value of \(A+B+C+8.\)
Want to solve more exponent puzzles? Try out the Exponents Warmup Quiz.
Here are some quick tips for strengthening your algebraic problem-solving skills:
- Ask yourself: what are the properties of the numbers and operations involved in this question?
- Read a question through in entirety before you start solving. For example, when calculating \(3333 \times 222 \times 11 \times 0\), if you start multiplying the large numbers as soon as you read them, then you feel pretty silly when you see the 0 at the end.
- Be on the look out for inverse operations and cancel these out before pulling out your calculator. For example, multiplying by 20 only to then divide by 5 is silly. Instead, just multiply by 4.
- Exponents trip a lot of people up. Don't just memorize the rules for working with exponents, but play around with each rule and build up your intuition.
- In an exponent tower such as \(2^{1^3},\) the order of operations is to evaluate from the top of the tower downward. So \(2^{1^3} = 2^{1} = 2\).
Additional articles on Brilliant that are related to this topic:
- Review the fundamental operations of arithmetic, and "PEMDAS," the order of operations. Completely mastering the basics will speed up your thought processes on harder problems.
- The backbone of Algebra is its use of variables and symbolic equations. Variable equations are one of the most generally applicable problem-solving tools in all of mathematics. Being able to simplify algebraic expressions, and to solve basic single and multivariable equations is considered a gateway skill for many other mathematical topics. And, of course, there are also exponential equations!
- Often, expressions can be simplified before they are solved, making the solving much easier. The distributive property and the rules for the exponential operation are particularly valuable for this.
- What is \(8^{-3}\) or \(8^{-\frac{1}{3}}\)? Kick up your understanding of exponents an extra few notches by generalizing to all rational exponents.