What are the three different ways to write 7^4 as a product of two powers? Explain why all three expressions are equal to 7^4.
To express 7^4 as a product of two powers, we can break it down in three different ways:
1) 7^4 = 7^2 * 7^2
In this expression, we are multiplying two powers with the same base (7) and adding their exponents (2 + 2 = 4). This is known as the product of powers property. Since the base (7) remains the same and the exponents (2 and 2) add up to the exponent of the original power (4), the expressions are equal.
2) 7^4 = (7^3) * 7^1
In this case, one of the powers has an exponent of 3 (7^3) and the other has an exponent of 1 (7^1). When we multiply these powers, we add their exponents (3 + 1 = 4). Again, the base (7) remains the same, and the exponents (3 and 1) together equal the exponent of the original power (4).
3) 7^4 = 7^5 / 7^1
In this expression, we divide two powers with the same base (7). When dividing powers with the same base, we subtract the exponent in the denominator from the exponent in the numerator (5 - 1 = 4). The base (7) stays the same, and the exponents (5 and 1) subtract to give us the exponent of the original power (4).
All three expressions are equal to 7^4 because they follow the rules of exponentiation, where the base remains the same and the exponents can be combined or manipulated according to various properties. These properties, such as the product of powers, allow us to express a power as a product of two powers.
To write 7^4 as a product of two powers, we can use the properties of exponents.
1. 7^4 can be written as (7^2)^2. This is because when we raise a power to another power, we multiply the exponents. In this case, 7^2 is raised to the power of 2, which gives us (49)^2 = 2401.
2. 7^4 can also be written as (7^3) × (7^1). This is because when we multiply powers with the same base, we add the exponents. In this case, 7^3 is raised to the power of 1, giving us 343 × 7 = 2401.
3. Finally, 7^4 can be written as (7^2) × (7^2). This is simply multiplying two powers with the same base. So, 7^2 × 7^2 = 49 × 49 = 2401.
All three expressions are equal to 7^4 because they use the properties of exponents. In the case of (7^2)^2, we are squaring 7^2, which results in the same value as 7^4. In the case of (7^3) × (7^1), we are multiplying two powers of 7, which again results in the same value as 7^4. Similarly, when we multiply (7^2) × (7^2), we are still multiplying two powers of 7, which gives us 7^4.