Evaluate (3(25)^(1/2))/2
The question asks to calculate the value of the expression given. It involves the following mathematical operations:
Taking the square root of 25, which is denoted by raising 25 to the power of 1/2.
Multiplying the result of that square root by 3.
Dividing the product obtained from step 2 by 2.
The expression can be read as "three times the square root of twenty-five, all over two." The question is looking for a numerical evaluation of this expression to find its simplified value.
$\frac{3 \left(\left(\right. 25 \left.\right)\right)^{\frac{1}{2}}}{2}$
Answer
Solution:
Step 1: Simplify the expression in the numerator.
Step 1.1: Express $25$ as a power of $5$, that is $5^2$. Thus, the expression becomes $\frac{3 \cdot (5^2)^{\frac{1}{2}}}{2}$.
Step 1.2: Utilize the property of exponents that states $(a^m)^n = a^{mn}$. The expression now simplifies to $\frac{3 \cdot 5^{2 \cdot \frac{1}{2}}}{2}$.
Step 1.3: Identify and eliminate common factors in the exponent.
Step 1.3.1: Remove the common factor from the exponent, resulting in $\frac{3 \cdot 5^{\cancel{2} \cdot \frac{1}{\cancel{2}}}}{2}$.
Step 1.3.2: Rewrite the simplified expression as $\frac{3 \cdot 5^1}{2}$.
Step 1.4: Calculate the value of the exponent, leading to $\frac{3 \cdot 5}{2}$.
Step 2: Perform the multiplication in the numerator.
- Multiply $3$ by $5$ to get $\frac{15}{2}$.
Step 3: Present the final result in various formats.
- Exact Form: $\frac{15}{2}$
- Decimal Form: $7.5$
- Mixed Number Form: $7 \frac{1}{2}$
Knowledge Notes:
Exponentiation: The operation of raising one number (the base) to the power of another (the exponent). For example, $a^n$ means multiplying $a$ by itself $n$ times.
Square Root: The square root of a number $x$ is a number $y$ such that $y^2 = x$. The square root of $25$ is $5$ because $5^2 = 25$.
Power Rule: For any nonzero number $a$ and integers $m$ and $n$, the power rule states that $(a^m)^n = a^{mn}$.
Simplification: The process of reducing an expression to its simplest form. This often involves combining like terms, applying exponent rules, and canceling common factors.
Rational Numbers: Numbers that can be expressed as the quotient or fraction $\frac{p}{q}$ of two integers, where $p$ is the numerator and $q$ is the nonzero denominator. The number $\frac{15}{2}$ is a rational number.
Decimal and Mixed Number Forms: Rational numbers can be expressed in decimal form by performing the division of the numerator by the denominator. They can also be written as mixed numbers if the numerator is greater than the denominator, which is a combination of a whole number and a proper fraction.
