Which expression is correctly developed to use the Product Rule of Exponents?

Question

(52)9
left parenthesis 5 squared right parenthesis superscript 9 baseline

327
32 superscript 7 baseline

108⋅108
10 superscript 8 baseline times 10 superscript 8 baseline

62⋅73

Answer 1

To correctly apply the Product Rule of Exponents, which states that \( a^m \cdot a^n = a^{m+n} \), we need to identify expressions that fall into the form of multiplication of two exponential terms with the same base.

Let's review the given expressions:

1. \((5^2)^9\): This expression is not directly using the Product Rule, but rather the Power of a Power Rule, which states that \( (a^m)^n = a^{m \cdot n} \).

2. \(32^7\): This is a single exponential term and does not involve products.

3. \(10^8 \cdot 10^8\): This is an example that fits the Product Rule because both terms have the same base (10). Applying the Product Rule: \[ 10^8 \cdot 10^8 = 10^{8+8} = 10^{16} \]

4. \(6^2 \cdot 7^3\): This expression combines two different bases (6 and 7) and does not fit the Product Rule of Exponents.

Thus, the expression that correctly utilizes the Product Rule of Exponents is: \(10^8 \cdot 10^8\).

Answer 2

can i get human numbers bro

Answer 3

Sure! Here are the expressions with the calculations written out in a more human-friendly manner:

1. (5²)⁹: This means \(5\) squared (which is \(25\)) raised to the power of \(9\). Using the power of a power rule, it becomes \(5^{2 \times 9} = 5^{18}\). Not the Product Rule.

2. 32⁷: This is just \(32\) raised to the power of \(7\). There's no multiplication of the same base here, so it doesn't involve the Product Rule.

3. 10⁸ × 10⁸: Here, you multiply \(10\) raised to the power of \(8\) by itself. Since these have the same base, you can add the exponents: \[ 10^{8 + 8} = 10^{16} \] This is an example of using the Product Rule.

4. 6² × 7³: This expression involves \(6\) raised to the power of \(2\) and \(7\) raised to the power of \(3\). Since the bases are different (6 and 7), it does not apply the Product Rule.

So, in simpler terms, the only expression that uses the Product Rule is 10⁸ × 10⁸, and it simplifies to 10¹⁶.

Answer 4

sooooooo.... 10 to the power of 16?

Answer 5

Yes, exactly! \(10^{16}\) means 10 raised to the power of 16. It represents the number:

\(10^{16} = 10,000,000,000,000,000\)

That’s 1 followed by 16 zeros!

Answer 6

and did you use the options i gave you

Answer 7

Yes, I used the options you provided. To clarify:

1. (5²)⁹: Not using the Product Rule; it uses the Power of a Power Rule.
2. 32⁷: Not using the Product Rule; it’s just a single exponent.
3. 10⁸ × 10⁸: This does use the Product Rule, which gives \(10^{8+8} = 10^{16}\).
4. 6² × 7³: Not using the Product Rule, as the bases are different.

So, among the options you gave, the expression that correctly applies the Product Rule is 10⁸ × 10⁸, and it simplifies to 10^{16}.