Evaluate 4(5)^4+24(5)^(1/3)
Brief Explanation of the Question:
This question is asking for the evaluation of a mathematical expression that includes both exponentiation and multiplication. The expression has two parts. The first part is 4 times 5 to the fourth power (4 * 5^4), and the second part is 24 times 5 to the power of one-third (24 * 5^(1/3)). The evaluator is expected to calculate the value of 5 raised to the fourth power and the cube root of 5 separately, then multiply each by their respective coefficients (4 and 24), and finally add those two products together to find the total sum.
$4 \left(\left(\right. 5 \left.\right)\right)^{4} + 24 \left(\left(\right. 5 \left.\right)\right)^{\frac{1}{3}}$
Answer
Solution:
Step 1:
Compute the value of $5$ raised to the $4^{th}$ power. $4 \times 625 + 24 \times 5^{\frac{1}{3}}$
Step 2:
Multiply the result from the previous step by $4$. $2500 + 24 \times 5^{\frac{1}{3}}$
Step 3:
Present the final result in both its exact and approximate decimal forms.
Exact Form: $2500 + 24 \times 5^{\frac{1}{3}}$ Decimal Form: Approximately $2541.03942272 \ldots$
Knowledge Notes:
To solve the given expression, we need to understand the following concepts:
Exponentiation: This is the process of raising a number to a power. In this case, $5^4$ means multiplying $5$ by itself $4$ times.
Fractional Exponents: A fractional exponent, such as $5^{\frac{1}{3}}$, represents a root. Specifically, $5^{\frac{1}{3}}$ is the cube root of $5$.
Order of Operations: When solving mathematical expressions, the order of operations must be followed. This usually means handling exponents before multiplication.
Exact vs. Decimal Form: The exact form of a number is the precise value, which can include roots and irrational numbers. The decimal form is an approximate value that is often easier to understand and use in calculations, but it may not be precise.
Multiplication: This is a basic arithmetic operation where a number is added to itself a certain number of times. For example, $4 \times 625$ is the same as adding $625$ to itself $4$ times.
By applying these concepts, we can solve the original problem step by step, starting with exponentiation, followed by multiplication, and finally presenting the result in both exact and decimal forms.
