Question

You rewrote the expression as 10
3
·5. Can you rewrite 10
3
·5 using a single base raised to a power? In other words, can you write this as a single power of 10, with no additional factors?
2
3
·5
4
= 2·2·2·5·5·5·5
= (2·5)·(2·5)·(2·5)·5
= 10·10·10·5
= 10
3
·5

Yes

No

Answer 1

Yes, you can rewrite \( 10^3 \cdot 5 \) using a single base raised to a power.

First, note that \( 10 = 2 \cdot 5 \). Therefore, we can express the \( 10^3 \) term in terms of its prime factors:

\[
10^3 = (2 \cdot 5)^3 = 2^3 \cdot 5^3
\]

Now substitute this back into the original expression:

\[
10^3 \cdot 5 = (2^3 \cdot 5^3) \cdot 5 = 2^3 \cdot 5^3 \cdot 5 = 2^3 \cdot 5^4
\]

To combine this into a single power of \( 10 \):

We know that:

\[
10^n = 2^n \cdot 5^n
\]

Now we need to express \( 2^3 \cdot 5^4 \):

This can be rewritten as:

\[
2^3 \cdot 5^4 = 2^3 \cdot 5^3 \cdot 5 = 10^3 \cdot 5
\]

Since you cannot simplify \( 2^3 \cdot 5^4 \) any further into a single base of \( 10 \) without additional factors. Therefore:

The expression \( 10^3 \cdot 5 \) cannot simplify down into a single power of 10 without extra factors.

So the answer is **No**, you cannot write \( 10^3 \cdot 5 \) as a single power of 10 without additional factors.