To simplify the expression [tex]\( x^{-\frac{4}{6}} \cdot x^{\frac{7}{8}} \)[/tex], we use the properties of exponents, specifically the property that [tex]\( a^m \cdot a^n = a^{m+n} \)[/tex]. This property allows us to combine the exponents when multiplying exponential terms with the same base.
First, we need to simplify each of the exponents:
- The exponent [tex]\(-\frac{4}{6}\)[/tex] can be simplified. Since [tex]\(-\frac{4}{6}\)[/tex] reduces to [tex]\(-\frac{2}{3}\)[/tex], we rewrite the expression [tex]\( x^{-\frac{4}{6}} \)[/tex] as [tex]\( x^{-\frac{2}{3}} \)[/tex].
Next, we sum the exponents:
- The simplified exponents are [tex]\(-\frac{2}{3}\)[/tex] and [tex]\(\frac{7}{8}\)[/tex].
- To add these fractions, we find a common denominator. The least common denominator (LCD) of 3 and 8 is 24.
Convert the exponents to have the same denominator:
- [tex]\(-\frac{2}{3}\)[/tex] converts to [tex]\(-\frac{16}{24}\)[/tex] because [tex]\(-\frac{2}{3} = -\frac{2 \times 8}{3 \times 8} = -\frac{16}{24}\)[/tex].
- [tex]\(\frac{7}{8}\)[/tex] converts to [tex]\(\frac{21}{24}\)[/tex] because [tex]\(\frac{7}{8} = \frac{7 \times 3}{8 \times 3} = \frac{21}{24}\)[/tex].
Now add the fractions:
- [tex]\(-\frac{16}{24} + \frac{21}{24} = \frac{-16 + 21}{24} = \frac{5}{24}\)[/tex].
Thus, the combined exponent is [tex]\(\frac{5}{24}\)[/tex].
Therefore, the simplified expression is:
[tex]\[ x^{\frac{5}{24}} \][/tex]
