Write 7 to the power of 4 in expanded form. Then check all that apply.
- The base is 7. – The exponent is 4. – The exponent tells you to multiply 7 together 4 times. – The exponent tells you to multiply 4 together 7 times. – The expanded form is 4×4×4×4×4×4×4. – The expanded form is 7×7×7×7.
The Correct Answer and Explanation is :
To write (7) raised to the power of (4) in expanded form, we express it as follows:
[
7^4 = 7 \times 7 \times 7 \times 7
]
Checking the Statements
- The base is 7.
True. In the expression (7^4), (7) is the base. - The exponent is 4.
True. The exponent in the expression (7^4) is (4). - The exponent tells you to multiply 7 together 4 times.
True. The exponent (4) indicates that the base (7) should be multiplied by itself a total of (4) times. - The exponent tells you to multiply 4 together 7 times.
False. The exponent does not indicate multiplying (4) together; it indicates how many times to use the base (7) in multiplication. - The expanded form is (4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4).
False. This expression represents (4^7), not (7^4). - The expanded form is (7 \times 7 \times 7 \times 7).
True. This is the correct expanded form of (7^4).
Explanation
Exponentiation is a mathematical operation that involves two numbers: a base and an exponent. The base is the number that is being multiplied, and the exponent indicates how many times the base is multiplied by itself. In this case, (7^4) means (7) is multiplied by itself (4) times. Therefore, the expression expands to (7 \times 7 \times 7 \times 7).
Understanding exponentiation is fundamental in mathematics because it simplifies the representation of large numbers and complex calculations. Instead of writing out many multiplications, we can use exponents to express them succinctly. For instance, (7^4) is much easier to write and comprehend than (7 \times 7 \times 7 \times 7).
The statements that are true highlight the basic principles of exponents, while the false statements show common misconceptions. It’s crucial to remember that the exponent applies to the base, not to the exponent itself or to other numbers unless explicitly stated. This understanding is essential for solving problems involving powers in algebra and beyond.