What expression is equivalent to 5 to the 6th power?
The Correct Answer and Explanation is:
The expression equivalent to (5^6) is:
[
5^6 = 5 \times 5 \times 5 \times 5 \times 5 \times 5
]
Explanation:
The notation (5^6) represents 5 raised to the power of 6, or 5 raised to the sixth exponent. In mathematical terms, an expression like (a^n) means that the base (a) is multiplied by itself (n) times. Here, the base is 5, and the exponent (or power) is 6. Therefore, (5^6) is the product of six 5’s multiplied together.
Step-by-Step Breakdown:
- Multiplication of 5’s: We can break down the expression (5^6) as follows:
[
5^6 = 5 \times 5 \times 5 \times 5 \times 5 \times 5
] - Simplification: To calculate the value of (5^6), you can group the multiplication in parts:
[
5 \times 5 = 25
]
Then:
[
25 \times 5 = 125
]
[
125 \times 5 = 625
]
[
625 \times 5 = 3125
]
[
3125 \times 5 = 15625
]
Thus, (5^6 = 15625).
Mathematical Rule of Exponents:
The exponentiation rule tells us that when the same base is multiplied by itself multiple times, we can simply write the product as a base raised to the power. For example:
[
a^m \times a^n = a^{m+n}
]
This rule is helpful for simplifying larger expressions, as seen in the example above.
Applications of Exponents:
Exponents, or powers, are widely used in various fields such as algebra, physics, computer science, and finance. They allow for a compact way of expressing large numbers, especially when calculating repeated multiplication, growth patterns, or exponential decay. In this case, (5^6) is a simple expression of repeated multiplication that can easily be expanded or calculated as shown.