Question Video: Simplifying an Expression Involving Multiplication of Integer Powers in the Real Numbers | Nagwa Question Video: Simplifying an Expression Involving Multiplication of Integer Powers in the Real Numbers | Nagwa

Question Video: Simplifying an Expression Involving Multiplication of Integer Powers in the Real Numbers Mathematics • Second Year of Preparatory School

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Simplify √2 × (√2)⁻² × (√2)⁻³.

02:42

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.

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