Video Transcript
Simplify the square root of two
multiplied by the square root of two to the power of negative two multiplied by the
square root of two to the power of negative three.
First, we recall the product rule
for exponents, which applies to factors with the same base. The rule states that the product of
𝑎 to the power of 𝑚 multiplied by 𝑎 to the power of 𝑛 is equal to 𝑎 to the
power of 𝑚 plus 𝑛 for any real number 𝑎 and integers 𝑚 and 𝑛. In our case, this means that the
square root of two multiplied by the square root of two to the power of negative two
multiplied by the square root of two to the power of negative three is equal to the
square root of two to the power of the sum of the three exponents.
We know that the square root of two
is equal to the same number to the first power. So, we have the square root of two
to the power of one plus negative two plus negative three. Then, we simplify the expression to
the square root of two to the power of negative four.
Now we recall the law of
multiplicative inverses, which states that 𝑎 to the power of 𝑛 multiplied by 𝑎 to
the power of negative 𝑛 is equal to one, where 𝑎 is a nonzero real number and 𝑛
is an integer. Since 𝑎 is nonzero, we can divide
both sides of the equation by 𝑎 to the power of 𝑛. This means that 𝑎 to the power of
negative 𝑛 is equal to one over 𝑎 to the power of 𝑛. For this problem, we have 𝑎 equal
to the square root of two and 𝑛 equal to four. Thus, the square root of two to the
power of negative four is equal to one over the square root of two to the power of
four.
We recall the property that states
that the square root of 𝑎 squared is equal to 𝑎, where 𝑎 is greater than
zero. So, we can write the square root of
two to the power of four as the product of powers of two using the product rule in
reverse. Using the property of the square
root squared, we know that the square root of two squared is equal to two, which
means our fraction now is one over two times two, which equals one-fourth.
In conclusion, the square root of
two multiplied by the square root of two to the power of negative two multiplied by
the square root of two to the power of negative three simplifies to one-fourth.
