Problem 35
Question
Copy and complete the statement. $$ \left(\frac{2}{7}\right)^{?}=\frac{8}{343} $$
Step-by-Step Solution
Verified Answer
The value of '?' that completes the statement is '3'
1Step 1: Recognize patterns
Look at the equation \(\left(\frac{2}{7}\right)^{?}=\frac{8}{343}\). Recognize that 8 and 343 are the cube of 2 and 7 respectively. This suggests that the unknown exponent is likely to be '3'.
2Step 2: Confirm the assumption
Confirm this by calculating \(\left(\frac{2}{7}\right)^{3}\). Simplifying this we get \(\frac{8}{343}\), which is the same as the right side of the original equation. Therefore the exponent '?' must be '3'.
3Step 3: Write the final statement
Substitute '?' with the discovered value 3. The completed statement will be then \(\left(\frac{2}{7}\right)^{3}=\frac{8}{343}\)
Key Concepts
Algebraic PatternsSimplifying ExpressionsRadicals and Rational Exponents
Algebraic Patterns
Algebraic patterns are fundamental building blocks within the world of algebra. They allow us to recognize relationships and structures in numerical data and expressions. For instance, consider the exercise involving exponentiation: \begin{quote}
Copy and complete the statement. $$ \left(\frac{2}{7}\right)^{?}=\frac{8}{343} $$
Simplifying Expressions
Simplifying expressions is a critical skill in algebra that involves modifying an equation or expression to make it easier to understand and work with. In the context of the provided exercise, simplifying the expression means confirming the pattern we have recognized.
After guessing that the exponent must be 3, we proceed to simplify the expression \begin{quote}\(\left(\frac{2}{7}\right)^{3}\)When simplified, we find that it indeed equals \begin{quote}\(\frac{8}{343}\)This simplification serves as confirmation that our initial pattern recognition led us to the correct solution. Sometimes simplifying expressions can involve factoring, expanding, combining like terms, or in this case, applying a suspected exponent to confirm an algebraic pattern. It's a process that, when mastered, can greatly aid in solving complex algebra problems with greater ease and accuracy.
After guessing that the exponent must be 3, we proceed to simplify the expression \begin{quote}\(\left(\frac{2}{7}\right)^{3}\)When simplified, we find that it indeed equals \begin{quote}\(\frac{8}{343}\)This simplification serves as confirmation that our initial pattern recognition led us to the correct solution. Sometimes simplifying expressions can involve factoring, expanding, combining like terms, or in this case, applying a suspected exponent to confirm an algebraic pattern. It's a process that, when mastered, can greatly aid in solving complex algebra problems with greater ease and accuracy.
Radicals and Rational Exponents
Radicals and rational exponents represent yet another nuanced concept within algebra. They are different ways to express the same mathematical operations—namely, roots and powers. The exercise we are examining:\begin{quote}
For example, knowing that the cube root of a number is the same as raising that number to the power of 1/3 allows students to toggle between the radical notation and the exponent notation, providing flexibility in how equations are approached and solved.
Copy and complete the statement. $$ \left(\frac{2}{7}\right)^{?}=\frac{8}{343} $$
is equivalent to taking the cube (third root) of 2/7. Understanding how to work with both radicals and rational exponents is a timesaver and can be a useful tool for simplifying complex algebraic expressions.For example, knowing that the cube root of a number is the same as raising that number to the power of 1/3 allows students to toggle between the radical notation and the exponent notation, providing flexibility in how equations are approached and solved.
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