Problem 18 Evaluate each exponential. $$ ... [FREE SOLUTION]

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Chapter 10: Problem 18

Evaluate each exponential. $$ 625^{1 / 4} $$

Short Answer

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Step by step solution

Understand the Exponential Expression

The expression to evaluate is $625^{1 / 4}$. This means finding the fourth root of 625.

Express 625 as a Power of a Number

Express 625 as a power of another number. Notice that $625 = 5^4$.

Substitute and Simplify

Rewrite the original expression using the new representation: $625^{1 / 4} = (5^4)^{1/4}$. By the properties of exponents, $(a^b)^c = a^{bc}$.

Apply the Exponent Rule

Simplify using the exponent rule: $(5^4)^{1/4} = 5^{4 \times 1/4} = 5^1 = 5$.

Write the Final Answer

The value of $625^{1 / 4}$ is 5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponents

An exponent is a mathematical notation that implies the number of times a number, called the base, is multiplied by itself. For example, in the expression $5^3$, the base is 5 and the exponent is 3, meaning $5 \times 5 \times 5$. Exponents are useful in representing large numbers more compactly and are written as a superscript to the right of the base number.

Roots

Roots are the inverse operation of exponents. They involve finding a number which, when raised to a specific power, gives the original number. For instance, in the exercise, we evaluated $625^{1/4}$. This means finding a number that, when raised to the 4th power, results in 625. The notation $x^{1/n}$ represents the n-th root of x. Common roots include the square root $\sqrt{x}$, or $x^{1/2}$, and the cube root $\sqrt[3]{x}$, or $x^{1/3}$.

Exponent Rules

Exponent rules simplify the manipulation of expressions involving exponents. Some key exponent rules include:

Product of Powers: $a^m \times a^n = a^{m+n}$. When multiplying with the same base, add the exponents.
Quotient of Powers: $a^m / a^n = a^{m-n}$. When dividing with the same base, subtract the exponents.
Power of a Power: $(a^m)^n = a^{m \times n}$. Raise the base to the product of the exponents.
Power of a Product: $(ab)^n = a^n b^n$. Distribute the exponent to both base terms.
Zero Exponent: $a^0 = 1$. Any non-zero base raised to the exponent zero equals one.
Negative Exponent: $a^{-n} = \frac{1}{a^n}$. A negative exponent indicates the reciprocal of the positive exponent value.

Using these rules, we can transform and simplify expressions, making it easier to solve problems like the one given in the exercise.

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91影视

Evaluate each exponential. $$ 625^{1 / 4} $$

Short Answer

Step by step solution

Understand the Exponential Expression

Express 625 as a Power of a Number

Substitute and Simplify

Apply the Exponent Rule

Write the Final Answer

Key Concepts

Exponents

Roots

Exponent Rules

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