Question

Simplify the expression. $$ \frac{6^{5 / 6}}{6^{1 / 6}} $$

Short answer

The simplified form of the expression \(\frac{6^{5 / 6}}{6^{1 / 6}}\) is \(6^{2/3}\).

Step-by-step solution

Identifying The Bases And Exponents

In the given expression, notice that both terms are powers of 6: \(6^{5 / 6}\) and \(6^{1 / 6}\). For both terms, 6 is the base and \(5/6\) and \(1/6\) are the powers.

Apply The Rule Of Exponents

The rule of exponents states that when we divide exponential terms with the same base, we subtract their exponents. This rule implies \(\frac{a^m}{a^n} = a^{m - n}\). For the given expression, the base is the same (which is 6) for the two terms. So, apply this rule to the expression like so: \(\frac{6^{5/6}}{6^{1/6}} = 6^{(5/6 - 1/6)}\).

Simplify The Exponent

Now simply subtract the fractions in the exponent: \( 5/6 - 1/6 = 4/6\). So, the expression simplifies to \(6^{4/6}\).

Reduce The Simplified Exponent

The exponent \(4/6\) reduces to \(2/3\) on simplification. So the final simplified form of the expression is \(6^{2/3}\).

Key concepts

Exponents and Powers

Exponents, also known as powers, are a shorthand way to represent repeated multiplication of the same number. The number being multiplied is called the base, and the number of times it is multiplied by itself is the exponent. For example, the expression

\( 6^{5/6} \)

can be understood as the base number 6 multiplied by itself 5/6 times, which may seem confusing initially. When the exponent is not a whole number, it indicates a root or a fractional power in arithmetic.

Understanding how to work with exponents is crucial for simplifying algebraic expressions efficiently. In the context of our exercise, we have a fractional exponent, which means we are dealing with both exponents and the concept of taking roots.

Rules of Exponents

When simplifying expressions involving exponents, there are several rules or laws that we follow to make the process systematic and reliable. One such rule is the 'Quotient of Powers' rule, which was applied in our textbook problem. This rule states:

\( \frac{a^m}{a^n} = a^{m-n} \)

For exponential terms with the same base, as seen in the expression given in our exercise \( \frac{6^{5/6}}{6^{1/6}} \), you subtract the exponent of the denominator from the exponent in the numerator. This rule allows you to simplify expressions without having to manually multiply each term.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operation symbols. In the given exercise,

\( \frac{6^{5/6}}{6^{1/6}} \)

is an algebraic expression that involves exponents with a common base. Simplifying algebraic expressions often involves combining like terms, which are terms that have the same variables raised to the same power, and using the rules of exponents. These processes transform the original complicated-looking expression into a simpler or more standard form without changing its value. In the textbook solution, we combined like terms by applying the appropriate rule of exponents.

Fractional Exponents

Fractional exponents can be intimidating, but they follow the same rules as integer exponents. A fractional exponent like \( 6^{2/3} \), which we obtained after simplifying our exercise, represents a power and a root. The numerator of the fraction tells us the power, and the denominator tells us the root. So, \( 6^{2/3} \) means the cube root of 6 squared.

Understanding fractional exponents is important because they often appear in higher-level algebra, calculus, and other mathematical fields. In our case, to fully grasp the final solution, comprehending that \( 6^{2/3} \) is an alternate way of writing \( \sqrt[3]{6^2} \) is essential.