Scientific notation

To arrange the values from least to greatest, we need a way to compare them on the same scale. A straightforward approach is to rewrite any scientific notation in ordinary decimal form so the sizes are easy to see.

Convert $X$ out of scientific notation:

$X=5.0\times 10^{-2}$

An exponent of $-2$ means shift the decimal point two places left:

1. First shift: $0.5$
2. Second shift: $0.05$

So, $X=0.05$.

Now compare the two decimals by lining them up:

$\begin{array}{rl}X& =0.050\\ Y& =0.055\end{array}$

They match in the tenths and hundredths places, but in the thousandths place $Y$ has $5$ while $X$ has $0$.

So, $X<Y$.

Convert $Z$ into decimal form:

$Z=4.9\times 10^{-1}$

An exponent of $-1$ means shift the decimal point one place left:

1. One shift: $0.49$

So, $Z=0.49$.

Using the decimal equivalents:

• $X=0.050$
• $Y=0.055$
• $Z=0.490$

Option A: $Z<X<Y⟹0.49<0.05<0.055$. Not true because $0.49$ is actually the greatest.

Option B: $Y<X<Z⟹0.055<0.05<0.49$. Not true because $0.055$ is bigger than $0.05$.

Option D: $X<Z<Y⟹0.05<0.49<0.055$. Not true because $0.49$ is bigger than $0.055$.

From the decimal comparisons, we have:

$0.050<0.055<0.490$

So the correct ordering is:

$X<Y<Z$

We want to put the given values in increasing order. Since some are in scientific notation, we can compare them efficiently by expressing them in a consistent form:

$\begin{array}{rl}X& =5.0\times 10^{-2}\\ Y& =0.055\\ Z& =4.9\times 10^{-1}\end{array}$

Convert $Y$ to scientific notation so it matches the style of $X$ and $Z$.

For $Y=0.055$, move the decimal two places right to make $5.5$. Because the decimal moved right 2 places, the power of ten is $-2$:

$Y=5.5\times 10^{-2}$

Compare the powers of ten:

• $X$ and $Y$ both use $10^{-2}$.
• $Z$ uses $10^{-1}$.

Because $-1>-2$, we know $10^{-1}$ represents a larger place value than $10^{-2}$. Therefore, $Z$ must be the greatest.

Since $X$ and $Y$ share the same power of ten ($10^{-2}$), compare just their leading numbers:

• Coefficient of $X=5.0$
• Coefficient of $Y=5.5$

Because $5.0<5.5$, it follows that $X<Y$.

We have found that $X<Y$ and that $Z$ is the largest.

• Option A cannot be right because it places $Z$ first (as if it were smallest).
• Option B cannot be right because it says $Y<X$.
• Option D cannot be right because it puts $Z$ before $Y$ (so $Z<Y$).

Putting the results together gives:

$X<Y<Z$

So the correct choice is Option C.

We are given the decimal $0.0000004$. This value is extremely tiny and clearly falls below 1. That observation connects directly to Scientific notation in your Number and Algebra Foundations work.

Scientific notation has the form $a\times 10^n$. Whether $n$ is positive or negative indicates the size of the original number:

• If $n$ is positive, the original number is at least 10.
• If $n$ is negative, the original number is a decimal between 0 and 1.

Because the problem states that $n$ is negative for $0.0000004$, it must be in that small-decimal interval.

Check each statement against the scientific-notation rule:

• If the number were equal to 1, then the exponent would be 0 because $1=1\times 10^0$.
• If the number were larger than 1, then the exponent would be 0 or positive, for example $400=4\times 10^2$.
• “exactly ten times” describes a single multiplication relationship, not whether a number lies between 0 and 1, so it doesn’t determine a negative exponent here.

Since $0.0000004$ is below 1, its scientific notation uses a negative exponent (here, $n=-7$) because we move the decimal point to the right to make a number from 1 to 10. So the exponent is negative because the number is smaller than 1.

To use elimination, we compare $0.0000004$ with $1$.

In scientific notation, the exponent $n$ counts how many places the decimal shifts. A negative $n$ happens when the number is between $0$ and $1$, which is part of Number and Algebra Foundations under Scientific notation.

Ask: is $0.0000004$ the same as $1$?

$0.0000004\ne 1$

So that relationship cannot be correct.

Next, check whether $0.0000004$ is greater than $1$.

Because it has 0 in the ones place and only tiny digits far to the right of the decimal point, it is far below one whole. So it is not larger than $1$.

Now test the “exactly ten times” claim.

If it were ten times $1$, it would be $10$. If $1$ were ten times the number, the number would be $0.1$. Neither equals $0.0000004$, so this option doesn’t work.

After eliminating the other choices, the remaining statement must be correct.

Since $0.0000004<1$, it is smaller than 1, which is why scientific notation uses a negative exponent (for example, $4\times 10^{-7}$) to show its size.